$lim_{ntoinfty} e^{-n} sum_{k=0}^{n} frac{n^k}{k!} = frac{1}{2}$ - basic methods











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Prove that $$limlimits_{ntoinfty} e^{-n} sum_{k=0}^{n} frac{n^k}{k!} = frac{1}{2}$$



This problem appeared on MSE many times, but each time it was solved using Poisson distribution or lots of integrals. I am wondering, is there any way to prove it using some basic properties of limits (their arithmetics, squeeze theorem etc.), definition of $e^x$ as $limlimits_{ntoinfty}(1+frac{x}{n})^n$, basic limits with $e$, binomial expansion and logarithms, but without using integrals, series, Stirling formula, asymptotics, Taylor series?



This problem was given to me by my mathematical analysis teacher, but it's not a homework, just additional problem to think on. My teacher claims it can be solved with knowledge introduced on lectures so far, which is not much, mainly things mentioned above. Of course, I can use theorems not mentioned on the lectures, but then I have to prove them, and again, with the baisc knowledge. I've been thinking about it for a few days and couldn't do any major progress in my attempts.



The answer by Sangchul Lee was written for the first version of this question which was not filled with details, for what I am sorry, and hence his answer does not answer my question.










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  • Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
    – Michael Lugo
    Nov 15 at 23:42










  • Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
    – Yves Daoust
    Nov 18 at 13:12








  • 1




    Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
    – Sangchul Lee
    Nov 18 at 19:16

















up vote
16
down vote

favorite
11












Prove that $$limlimits_{ntoinfty} e^{-n} sum_{k=0}^{n} frac{n^k}{k!} = frac{1}{2}$$



This problem appeared on MSE many times, but each time it was solved using Poisson distribution or lots of integrals. I am wondering, is there any way to prove it using some basic properties of limits (their arithmetics, squeeze theorem etc.), definition of $e^x$ as $limlimits_{ntoinfty}(1+frac{x}{n})^n$, basic limits with $e$, binomial expansion and logarithms, but without using integrals, series, Stirling formula, asymptotics, Taylor series?



This problem was given to me by my mathematical analysis teacher, but it's not a homework, just additional problem to think on. My teacher claims it can be solved with knowledge introduced on lectures so far, which is not much, mainly things mentioned above. Of course, I can use theorems not mentioned on the lectures, but then I have to prove them, and again, with the baisc knowledge. I've been thinking about it for a few days and couldn't do any major progress in my attempts.



The answer by Sangchul Lee was written for the first version of this question which was not filled with details, for what I am sorry, and hence his answer does not answer my question.










share|cite|improve this question

















This question has an open bounty worth +50
reputation from user128409235 ending tomorrow.


This question has not received enough attention.
















  • Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
    – Michael Lugo
    Nov 15 at 23:42










  • Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
    – Yves Daoust
    Nov 18 at 13:12








  • 1




    Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
    – Sangchul Lee
    Nov 18 at 19:16















up vote
16
down vote

favorite
11









up vote
16
down vote

favorite
11






11





Prove that $$limlimits_{ntoinfty} e^{-n} sum_{k=0}^{n} frac{n^k}{k!} = frac{1}{2}$$



This problem appeared on MSE many times, but each time it was solved using Poisson distribution or lots of integrals. I am wondering, is there any way to prove it using some basic properties of limits (their arithmetics, squeeze theorem etc.), definition of $e^x$ as $limlimits_{ntoinfty}(1+frac{x}{n})^n$, basic limits with $e$, binomial expansion and logarithms, but without using integrals, series, Stirling formula, asymptotics, Taylor series?



This problem was given to me by my mathematical analysis teacher, but it's not a homework, just additional problem to think on. My teacher claims it can be solved with knowledge introduced on lectures so far, which is not much, mainly things mentioned above. Of course, I can use theorems not mentioned on the lectures, but then I have to prove them, and again, with the baisc knowledge. I've been thinking about it for a few days and couldn't do any major progress in my attempts.



The answer by Sangchul Lee was written for the first version of this question which was not filled with details, for what I am sorry, and hence his answer does not answer my question.










share|cite|improve this question















Prove that $$limlimits_{ntoinfty} e^{-n} sum_{k=0}^{n} frac{n^k}{k!} = frac{1}{2}$$



This problem appeared on MSE many times, but each time it was solved using Poisson distribution or lots of integrals. I am wondering, is there any way to prove it using some basic properties of limits (their arithmetics, squeeze theorem etc.), definition of $e^x$ as $limlimits_{ntoinfty}(1+frac{x}{n})^n$, basic limits with $e$, binomial expansion and logarithms, but without using integrals, series, Stirling formula, asymptotics, Taylor series?



This problem was given to me by my mathematical analysis teacher, but it's not a homework, just additional problem to think on. My teacher claims it can be solved with knowledge introduced on lectures so far, which is not much, mainly things mentioned above. Of course, I can use theorems not mentioned on the lectures, but then I have to prove them, and again, with the baisc knowledge. I've been thinking about it for a few days and couldn't do any major progress in my attempts.



The answer by Sangchul Lee was written for the first version of this question which was not filled with details, for what I am sorry, and hence his answer does not answer my question.







calculus real-analysis limits eulers-constant






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edited Nov 20 at 12:29









leonbloy

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39.6k645105










asked Nov 15 at 22:43









user128409235

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This question has an open bounty worth +50
reputation from user128409235 ending tomorrow.


This question has not received enough attention.








This question has an open bounty worth +50
reputation from user128409235 ending tomorrow.


This question has not received enough attention.














  • Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
    – Michael Lugo
    Nov 15 at 23:42










  • Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
    – Yves Daoust
    Nov 18 at 13:12








  • 1




    Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
    – Sangchul Lee
    Nov 18 at 19:16




















  • Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
    – Michael Lugo
    Nov 15 at 23:42










  • Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
    – Yves Daoust
    Nov 18 at 13:12








  • 1




    Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
    – Sangchul Lee
    Nov 18 at 19:16


















Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
– Michael Lugo
Nov 15 at 23:42




Previous times this question has been asked: math.stackexchange.com/questions/2603315/…, math.stackexchange.com/questions/160248/… . I'm not flagging as duplicate because the question specifically asks for basic methods.
– Michael Lugo
Nov 15 at 23:42












Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
– Yves Daoust
Nov 18 at 13:12






Interestingly, the general term $n^k/k!$ is maximum when $k=n$. And by some magic, the sum "before" equals the sum "after" (asymptotically).
– Yves Daoust
Nov 18 at 13:12






1




1




Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
– Sangchul Lee
Nov 18 at 19:16






Without an algebraic miracle, I do not think there is a simple and rigorous solution. My rationale is that any solution should demonstrate the understanding on the concentration behavior of the function $k mapsto frac{n^k}{k!}e^{-n}$ around $k = n$. Both the probabilistic solution (using Poisson distribution + CLT) and the solution using integral show this. My solution is also based on this kind of observation.
– Sangchul Lee
Nov 18 at 19:16












2 Answers
2






active

oldest

votes

















up vote
24
down vote













Heuristic argument. Although far from being rigorous, one can provide a heuristic computation which explains why we expect the answer to be $frac{1}{2}$. Notice that



$$ frac{n^{n+j}/(n+j)!}{n^n / n!} = begin{cases}
prod_{k=1}^{j} frac{n}{n+k}, & j geq 1 \
1, & j = 0, -1 \
prod_{k=1}^{-j-1} frac{n-k}{n}, & j leq -2
end{cases}
$$



In any of cases, taking logarithm and applying the approximation $log(1+x) approx x$ shows that the above quantity is approximately $e^{-j^2/2n}$. So



$$
e^{-n} sum_{k=0}^{n} frac{n^k}{k!}
= frac{sum_{j=-n}^{0} frac{n!}{n^n sqrt{n}} frac{n^{n+j}}{(n+j)!} }{sum_{j=-n}^{infty} frac{n!}{n^n sqrt{n}}frac{n^{n+j}}{(n+j)!} }
approx frac{sum_{j=-n}^{0} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }{sum_{j=-n}^{infty} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }
approx frac{int_{-infty}^{0} e^{-x^2/2} , dx}{int_{-infty}^{infty} e^{-x^2/2} , dx}
= frac{1}{2}.
$$



Every solution that I know is more or less a rigorous realization of this kind of observation, and so, it is either involved or requiring extra knowledge.





Elementary proof. Define $A_n$, $B_n$ and $C_n$ by



$$ A_n := e^{-n} sum_{k=0}^{n} frac{n^k}{k!}, qquad B_n := e^{-n} sum_{k=n+1}^{infty} frac{n^k}{k!}, qquad C_n = frac{n^{n} e^{-n}}{n!}. $$



From the Taylor expansion of the exponential function, we know that $A_n + B_n = 1$. In order to show the desired convergence, it suffices to show that the following claim holds.




Claim. $A_n/B_n to 1$ as $ntoinfty$.




Indeed, once this is proved, then both $A_n$ and $B_n$ converge to $frac{1}{2}$ as $ntoinfty$.



Proof of Claim. Using the substitution $k = n-j$ followed by $p = j-1$, one may write



begin{align*}
frac{A_n}{C_n}
&= sum_{j=0}^{n} frac{n!}{(n-j)!n^j}
= 2 + sum_{p=1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right).
end{align*}



Similarly, applying the substitution $k = n+p$, one may write



begin{align*}
frac{B_n}{C_n}
&= sum_{p=1}^{infty} frac{n!n^p}{(n+p)!}
= sum_{p=1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right).
end{align*}



1. Estimation of leading sums. Fix $alpha in left( 0, frac{1}{3} right)$ and set $N = N(alpha) = leftlceil n^{(1+alpha)/2} rightrceil$. Then using the asymptotic formula $1+x = e^{x+mathcal{O}(x^2)}$ as $x to 0$, for $1 leq p leq N$ we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
= expleft{ -frac{p^2}{2n} + mathcal{O}left( n^{-(1-3alpha)/2} right) right}
= prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right). $$



In particular, there exists a constant $C > 0$, depending only on $alpha$, such that



$$ maxBigg{ prod_{l=1}^{N} left( 1 - frac{l}{n} right), prod_{l=1}^{N} left( frac{1}{1 + frac{l}{n}} right) Bigg} leq C e^{-frac{1}{2}n^{alpha}}. $$



2. Estimation of tail sums. In this time, consider $p > N$. Then we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq C e^{-frac{1}{2}n^{alpha}} left( 1 - frac{N}{n} right)^{p-N}
quad text{and} quad
prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq C e^{-frac{1}{2}n^{alpha}} left( frac{1}{1 + frac{N}{n}} right)^{p-N}. $$



So the tail sum for $A_n/C_n$ can be bounded from above by



$$ sum_{p=N+1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{n} right)^k
leq frac{Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}, $$



and likewise,



$$ sum_{p=N+1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{N+n} right)^k
leq frac{2Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq 2Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}. $$



3. Conclusion. Combining altogether,



$$ frac{A_n}{B_n} = frac{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + mathcal{O}(1)}{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + o(1)}, $$



which can be easily shown to converge to $1$ as $ntoinfty$, since $sum_{p=1}^{N} e^{-frac{p^2}{2n}} geq sqrt{n} , e^{-1/2} to infty$ as $ntoinfty$. (In fact, this sum is $(1+o(1))sqrt{pi n/2}$ as $ntoinfty$.)






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  • Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
    – user128409235
    Nov 18 at 9:02


















up vote
0
down vote













Only a comment.



I also use Stirling and (a Riemann) integral, but in a short way as a rough method.



$displaystyle e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} $



$displaystyle asymp e^{-n}sumlimits_{k=1}^{n-1}frac{e^k e^{n-k}}{sqrt{2pi k (1+mathcal{O}(1/k))}sqrt{2pi (n-k)(1+mathcal{O}(1/(n-k)))}} $



$displaystyle asymp frac{1}{2pi}frac{1}{n}sumlimits_{k=1}^{n-1}frac{1}{sqrt{frac{k}{n}left(1-frac{k}{n}right)}} asymp frac{1}{2pi} intlimits_0^1frac{dx}{sqrt{x(1-x)}} = frac{1}{2}$



Note: I use $asymp$ for "same asymptotically" in terms of $ntoinfty$ .



Maybe someone has an idea, how to simplify that, so that $pi$ and the integral disappear ?!






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  • How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
    – Diger
    2 hours ago













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2 Answers
2






active

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2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
24
down vote













Heuristic argument. Although far from being rigorous, one can provide a heuristic computation which explains why we expect the answer to be $frac{1}{2}$. Notice that



$$ frac{n^{n+j}/(n+j)!}{n^n / n!} = begin{cases}
prod_{k=1}^{j} frac{n}{n+k}, & j geq 1 \
1, & j = 0, -1 \
prod_{k=1}^{-j-1} frac{n-k}{n}, & j leq -2
end{cases}
$$



In any of cases, taking logarithm and applying the approximation $log(1+x) approx x$ shows that the above quantity is approximately $e^{-j^2/2n}$. So



$$
e^{-n} sum_{k=0}^{n} frac{n^k}{k!}
= frac{sum_{j=-n}^{0} frac{n!}{n^n sqrt{n}} frac{n^{n+j}}{(n+j)!} }{sum_{j=-n}^{infty} frac{n!}{n^n sqrt{n}}frac{n^{n+j}}{(n+j)!} }
approx frac{sum_{j=-n}^{0} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }{sum_{j=-n}^{infty} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }
approx frac{int_{-infty}^{0} e^{-x^2/2} , dx}{int_{-infty}^{infty} e^{-x^2/2} , dx}
= frac{1}{2}.
$$



Every solution that I know is more or less a rigorous realization of this kind of observation, and so, it is either involved or requiring extra knowledge.





Elementary proof. Define $A_n$, $B_n$ and $C_n$ by



$$ A_n := e^{-n} sum_{k=0}^{n} frac{n^k}{k!}, qquad B_n := e^{-n} sum_{k=n+1}^{infty} frac{n^k}{k!}, qquad C_n = frac{n^{n} e^{-n}}{n!}. $$



From the Taylor expansion of the exponential function, we know that $A_n + B_n = 1$. In order to show the desired convergence, it suffices to show that the following claim holds.




Claim. $A_n/B_n to 1$ as $ntoinfty$.




Indeed, once this is proved, then both $A_n$ and $B_n$ converge to $frac{1}{2}$ as $ntoinfty$.



Proof of Claim. Using the substitution $k = n-j$ followed by $p = j-1$, one may write



begin{align*}
frac{A_n}{C_n}
&= sum_{j=0}^{n} frac{n!}{(n-j)!n^j}
= 2 + sum_{p=1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right).
end{align*}



Similarly, applying the substitution $k = n+p$, one may write



begin{align*}
frac{B_n}{C_n}
&= sum_{p=1}^{infty} frac{n!n^p}{(n+p)!}
= sum_{p=1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right).
end{align*}



1. Estimation of leading sums. Fix $alpha in left( 0, frac{1}{3} right)$ and set $N = N(alpha) = leftlceil n^{(1+alpha)/2} rightrceil$. Then using the asymptotic formula $1+x = e^{x+mathcal{O}(x^2)}$ as $x to 0$, for $1 leq p leq N$ we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
= expleft{ -frac{p^2}{2n} + mathcal{O}left( n^{-(1-3alpha)/2} right) right}
= prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right). $$



In particular, there exists a constant $C > 0$, depending only on $alpha$, such that



$$ maxBigg{ prod_{l=1}^{N} left( 1 - frac{l}{n} right), prod_{l=1}^{N} left( frac{1}{1 + frac{l}{n}} right) Bigg} leq C e^{-frac{1}{2}n^{alpha}}. $$



2. Estimation of tail sums. In this time, consider $p > N$. Then we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq C e^{-frac{1}{2}n^{alpha}} left( 1 - frac{N}{n} right)^{p-N}
quad text{and} quad
prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq C e^{-frac{1}{2}n^{alpha}} left( frac{1}{1 + frac{N}{n}} right)^{p-N}. $$



So the tail sum for $A_n/C_n$ can be bounded from above by



$$ sum_{p=N+1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{n} right)^k
leq frac{Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}, $$



and likewise,



$$ sum_{p=N+1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{N+n} right)^k
leq frac{2Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq 2Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}. $$



3. Conclusion. Combining altogether,



$$ frac{A_n}{B_n} = frac{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + mathcal{O}(1)}{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + o(1)}, $$



which can be easily shown to converge to $1$ as $ntoinfty$, since $sum_{p=1}^{N} e^{-frac{p^2}{2n}} geq sqrt{n} , e^{-1/2} to infty$ as $ntoinfty$. (In fact, this sum is $(1+o(1))sqrt{pi n/2}$ as $ntoinfty$.)






share|cite|improve this answer























  • Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
    – user128409235
    Nov 18 at 9:02















up vote
24
down vote













Heuristic argument. Although far from being rigorous, one can provide a heuristic computation which explains why we expect the answer to be $frac{1}{2}$. Notice that



$$ frac{n^{n+j}/(n+j)!}{n^n / n!} = begin{cases}
prod_{k=1}^{j} frac{n}{n+k}, & j geq 1 \
1, & j = 0, -1 \
prod_{k=1}^{-j-1} frac{n-k}{n}, & j leq -2
end{cases}
$$



In any of cases, taking logarithm and applying the approximation $log(1+x) approx x$ shows that the above quantity is approximately $e^{-j^2/2n}$. So



$$
e^{-n} sum_{k=0}^{n} frac{n^k}{k!}
= frac{sum_{j=-n}^{0} frac{n!}{n^n sqrt{n}} frac{n^{n+j}}{(n+j)!} }{sum_{j=-n}^{infty} frac{n!}{n^n sqrt{n}}frac{n^{n+j}}{(n+j)!} }
approx frac{sum_{j=-n}^{0} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }{sum_{j=-n}^{infty} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }
approx frac{int_{-infty}^{0} e^{-x^2/2} , dx}{int_{-infty}^{infty} e^{-x^2/2} , dx}
= frac{1}{2}.
$$



Every solution that I know is more or less a rigorous realization of this kind of observation, and so, it is either involved or requiring extra knowledge.





Elementary proof. Define $A_n$, $B_n$ and $C_n$ by



$$ A_n := e^{-n} sum_{k=0}^{n} frac{n^k}{k!}, qquad B_n := e^{-n} sum_{k=n+1}^{infty} frac{n^k}{k!}, qquad C_n = frac{n^{n} e^{-n}}{n!}. $$



From the Taylor expansion of the exponential function, we know that $A_n + B_n = 1$. In order to show the desired convergence, it suffices to show that the following claim holds.




Claim. $A_n/B_n to 1$ as $ntoinfty$.




Indeed, once this is proved, then both $A_n$ and $B_n$ converge to $frac{1}{2}$ as $ntoinfty$.



Proof of Claim. Using the substitution $k = n-j$ followed by $p = j-1$, one may write



begin{align*}
frac{A_n}{C_n}
&= sum_{j=0}^{n} frac{n!}{(n-j)!n^j}
= 2 + sum_{p=1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right).
end{align*}



Similarly, applying the substitution $k = n+p$, one may write



begin{align*}
frac{B_n}{C_n}
&= sum_{p=1}^{infty} frac{n!n^p}{(n+p)!}
= sum_{p=1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right).
end{align*}



1. Estimation of leading sums. Fix $alpha in left( 0, frac{1}{3} right)$ and set $N = N(alpha) = leftlceil n^{(1+alpha)/2} rightrceil$. Then using the asymptotic formula $1+x = e^{x+mathcal{O}(x^2)}$ as $x to 0$, for $1 leq p leq N$ we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
= expleft{ -frac{p^2}{2n} + mathcal{O}left( n^{-(1-3alpha)/2} right) right}
= prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right). $$



In particular, there exists a constant $C > 0$, depending only on $alpha$, such that



$$ maxBigg{ prod_{l=1}^{N} left( 1 - frac{l}{n} right), prod_{l=1}^{N} left( frac{1}{1 + frac{l}{n}} right) Bigg} leq C e^{-frac{1}{2}n^{alpha}}. $$



2. Estimation of tail sums. In this time, consider $p > N$. Then we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq C e^{-frac{1}{2}n^{alpha}} left( 1 - frac{N}{n} right)^{p-N}
quad text{and} quad
prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq C e^{-frac{1}{2}n^{alpha}} left( frac{1}{1 + frac{N}{n}} right)^{p-N}. $$



So the tail sum for $A_n/C_n$ can be bounded from above by



$$ sum_{p=N+1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{n} right)^k
leq frac{Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}, $$



and likewise,



$$ sum_{p=N+1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{N+n} right)^k
leq frac{2Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq 2Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}. $$



3. Conclusion. Combining altogether,



$$ frac{A_n}{B_n} = frac{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + mathcal{O}(1)}{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + o(1)}, $$



which can be easily shown to converge to $1$ as $ntoinfty$, since $sum_{p=1}^{N} e^{-frac{p^2}{2n}} geq sqrt{n} , e^{-1/2} to infty$ as $ntoinfty$. (In fact, this sum is $(1+o(1))sqrt{pi n/2}$ as $ntoinfty$.)






share|cite|improve this answer























  • Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
    – user128409235
    Nov 18 at 9:02













up vote
24
down vote










up vote
24
down vote









Heuristic argument. Although far from being rigorous, one can provide a heuristic computation which explains why we expect the answer to be $frac{1}{2}$. Notice that



$$ frac{n^{n+j}/(n+j)!}{n^n / n!} = begin{cases}
prod_{k=1}^{j} frac{n}{n+k}, & j geq 1 \
1, & j = 0, -1 \
prod_{k=1}^{-j-1} frac{n-k}{n}, & j leq -2
end{cases}
$$



In any of cases, taking logarithm and applying the approximation $log(1+x) approx x$ shows that the above quantity is approximately $e^{-j^2/2n}$. So



$$
e^{-n} sum_{k=0}^{n} frac{n^k}{k!}
= frac{sum_{j=-n}^{0} frac{n!}{n^n sqrt{n}} frac{n^{n+j}}{(n+j)!} }{sum_{j=-n}^{infty} frac{n!}{n^n sqrt{n}}frac{n^{n+j}}{(n+j)!} }
approx frac{sum_{j=-n}^{0} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }{sum_{j=-n}^{infty} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }
approx frac{int_{-infty}^{0} e^{-x^2/2} , dx}{int_{-infty}^{infty} e^{-x^2/2} , dx}
= frac{1}{2}.
$$



Every solution that I know is more or less a rigorous realization of this kind of observation, and so, it is either involved or requiring extra knowledge.





Elementary proof. Define $A_n$, $B_n$ and $C_n$ by



$$ A_n := e^{-n} sum_{k=0}^{n} frac{n^k}{k!}, qquad B_n := e^{-n} sum_{k=n+1}^{infty} frac{n^k}{k!}, qquad C_n = frac{n^{n} e^{-n}}{n!}. $$



From the Taylor expansion of the exponential function, we know that $A_n + B_n = 1$. In order to show the desired convergence, it suffices to show that the following claim holds.




Claim. $A_n/B_n to 1$ as $ntoinfty$.




Indeed, once this is proved, then both $A_n$ and $B_n$ converge to $frac{1}{2}$ as $ntoinfty$.



Proof of Claim. Using the substitution $k = n-j$ followed by $p = j-1$, one may write



begin{align*}
frac{A_n}{C_n}
&= sum_{j=0}^{n} frac{n!}{(n-j)!n^j}
= 2 + sum_{p=1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right).
end{align*}



Similarly, applying the substitution $k = n+p$, one may write



begin{align*}
frac{B_n}{C_n}
&= sum_{p=1}^{infty} frac{n!n^p}{(n+p)!}
= sum_{p=1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right).
end{align*}



1. Estimation of leading sums. Fix $alpha in left( 0, frac{1}{3} right)$ and set $N = N(alpha) = leftlceil n^{(1+alpha)/2} rightrceil$. Then using the asymptotic formula $1+x = e^{x+mathcal{O}(x^2)}$ as $x to 0$, for $1 leq p leq N$ we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
= expleft{ -frac{p^2}{2n} + mathcal{O}left( n^{-(1-3alpha)/2} right) right}
= prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right). $$



In particular, there exists a constant $C > 0$, depending only on $alpha$, such that



$$ maxBigg{ prod_{l=1}^{N} left( 1 - frac{l}{n} right), prod_{l=1}^{N} left( frac{1}{1 + frac{l}{n}} right) Bigg} leq C e^{-frac{1}{2}n^{alpha}}. $$



2. Estimation of tail sums. In this time, consider $p > N$. Then we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq C e^{-frac{1}{2}n^{alpha}} left( 1 - frac{N}{n} right)^{p-N}
quad text{and} quad
prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq C e^{-frac{1}{2}n^{alpha}} left( frac{1}{1 + frac{N}{n}} right)^{p-N}. $$



So the tail sum for $A_n/C_n$ can be bounded from above by



$$ sum_{p=N+1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{n} right)^k
leq frac{Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}, $$



and likewise,



$$ sum_{p=N+1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{N+n} right)^k
leq frac{2Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq 2Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}. $$



3. Conclusion. Combining altogether,



$$ frac{A_n}{B_n} = frac{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + mathcal{O}(1)}{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + o(1)}, $$



which can be easily shown to converge to $1$ as $ntoinfty$, since $sum_{p=1}^{N} e^{-frac{p^2}{2n}} geq sqrt{n} , e^{-1/2} to infty$ as $ntoinfty$. (In fact, this sum is $(1+o(1))sqrt{pi n/2}$ as $ntoinfty$.)






share|cite|improve this answer














Heuristic argument. Although far from being rigorous, one can provide a heuristic computation which explains why we expect the answer to be $frac{1}{2}$. Notice that



$$ frac{n^{n+j}/(n+j)!}{n^n / n!} = begin{cases}
prod_{k=1}^{j} frac{n}{n+k}, & j geq 1 \
1, & j = 0, -1 \
prod_{k=1}^{-j-1} frac{n-k}{n}, & j leq -2
end{cases}
$$



In any of cases, taking logarithm and applying the approximation $log(1+x) approx x$ shows that the above quantity is approximately $e^{-j^2/2n}$. So



$$
e^{-n} sum_{k=0}^{n} frac{n^k}{k!}
= frac{sum_{j=-n}^{0} frac{n!}{n^n sqrt{n}} frac{n^{n+j}}{(n+j)!} }{sum_{j=-n}^{infty} frac{n!}{n^n sqrt{n}}frac{n^{n+j}}{(n+j)!} }
approx frac{sum_{j=-n}^{0} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }{sum_{j=-n}^{infty} e^{-(j/sqrt{n})^2/2} frac{1}{sqrt{n}} }
approx frac{int_{-infty}^{0} e^{-x^2/2} , dx}{int_{-infty}^{infty} e^{-x^2/2} , dx}
= frac{1}{2}.
$$



Every solution that I know is more or less a rigorous realization of this kind of observation, and so, it is either involved or requiring extra knowledge.





Elementary proof. Define $A_n$, $B_n$ and $C_n$ by



$$ A_n := e^{-n} sum_{k=0}^{n} frac{n^k}{k!}, qquad B_n := e^{-n} sum_{k=n+1}^{infty} frac{n^k}{k!}, qquad C_n = frac{n^{n} e^{-n}}{n!}. $$



From the Taylor expansion of the exponential function, we know that $A_n + B_n = 1$. In order to show the desired convergence, it suffices to show that the following claim holds.




Claim. $A_n/B_n to 1$ as $ntoinfty$.




Indeed, once this is proved, then both $A_n$ and $B_n$ converge to $frac{1}{2}$ as $ntoinfty$.



Proof of Claim. Using the substitution $k = n-j$ followed by $p = j-1$, one may write



begin{align*}
frac{A_n}{C_n}
&= sum_{j=0}^{n} frac{n!}{(n-j)!n^j}
= 2 + sum_{p=1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right).
end{align*}



Similarly, applying the substitution $k = n+p$, one may write



begin{align*}
frac{B_n}{C_n}
&= sum_{p=1}^{infty} frac{n!n^p}{(n+p)!}
= sum_{p=1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right).
end{align*}



1. Estimation of leading sums. Fix $alpha in left( 0, frac{1}{3} right)$ and set $N = N(alpha) = leftlceil n^{(1+alpha)/2} rightrceil$. Then using the asymptotic formula $1+x = e^{x+mathcal{O}(x^2)}$ as $x to 0$, for $1 leq p leq N$ we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
= expleft{ -frac{p^2}{2n} + mathcal{O}left( n^{-(1-3alpha)/2} right) right}
= prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right). $$



In particular, there exists a constant $C > 0$, depending only on $alpha$, such that



$$ maxBigg{ prod_{l=1}^{N} left( 1 - frac{l}{n} right), prod_{l=1}^{N} left( frac{1}{1 + frac{l}{n}} right) Bigg} leq C e^{-frac{1}{2}n^{alpha}}. $$



2. Estimation of tail sums. In this time, consider $p > N$. Then we have



$$ prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq C e^{-frac{1}{2}n^{alpha}} left( 1 - frac{N}{n} right)^{p-N}
quad text{and} quad
prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq C e^{-frac{1}{2}n^{alpha}} left( frac{1}{1 + frac{N}{n}} right)^{p-N}. $$



So the tail sum for $A_n/C_n$ can be bounded from above by



$$ sum_{p=N+1}^{n-1} prod_{l=1}^{p} left( 1 - frac{l}{n} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{n} right)^k
leq frac{Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}, $$



and likewise,



$$ sum_{p=N+1}^{infty} prod_{l=1}^{p} left( frac{1}{1 + frac{l}{n}} right)
leq Ce^{-frac{1}{2}n^{alpha}} sum_{k = 0}^{infty} left( 1 - frac{N}{N+n} right)^k
leq frac{2Cn}{N} e^{-frac{1}{2}n^{alpha}}
leq 2Cn^{(1-alpha)/2} e^{-frac{1}{2}n^{alpha}}. $$



3. Conclusion. Combining altogether,



$$ frac{A_n}{B_n} = frac{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + mathcal{O}(1)}{left( 1 + o(1) right) sum_{p=1}^{N} e^{-frac{p^2}{2n}} + o(1)}, $$



which can be easily shown to converge to $1$ as $ntoinfty$, since $sum_{p=1}^{N} e^{-frac{p^2}{2n}} geq sqrt{n} , e^{-1/2} to infty$ as $ntoinfty$. (In fact, this sum is $(1+o(1))sqrt{pi n/2}$ as $ntoinfty$.)







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 19 at 23:10

























answered Nov 16 at 2:30









Sangchul Lee

89.8k12161262




89.8k12161262












  • Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
    – user128409235
    Nov 18 at 9:02


















  • Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
    – user128409235
    Nov 18 at 9:02
















Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
– user128409235
Nov 18 at 9:02




Thank you for your answer! I've filled my question with more details to be more specific on the kind of solution I am looking form. Unfortunately, yours doesn't fall in this category, but for your efforts, I am giving you +1.
– user128409235
Nov 18 at 9:02










up vote
0
down vote













Only a comment.



I also use Stirling and (a Riemann) integral, but in a short way as a rough method.



$displaystyle e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} $



$displaystyle asymp e^{-n}sumlimits_{k=1}^{n-1}frac{e^k e^{n-k}}{sqrt{2pi k (1+mathcal{O}(1/k))}sqrt{2pi (n-k)(1+mathcal{O}(1/(n-k)))}} $



$displaystyle asymp frac{1}{2pi}frac{1}{n}sumlimits_{k=1}^{n-1}frac{1}{sqrt{frac{k}{n}left(1-frac{k}{n}right)}} asymp frac{1}{2pi} intlimits_0^1frac{dx}{sqrt{x(1-x)}} = frac{1}{2}$



Note: I use $asymp$ for "same asymptotically" in terms of $ntoinfty$ .



Maybe someone has an idea, how to simplify that, so that $pi$ and the integral disappear ?!






share|cite|improve this answer























  • How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
    – Diger
    2 hours ago

















up vote
0
down vote













Only a comment.



I also use Stirling and (a Riemann) integral, but in a short way as a rough method.



$displaystyle e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} $



$displaystyle asymp e^{-n}sumlimits_{k=1}^{n-1}frac{e^k e^{n-k}}{sqrt{2pi k (1+mathcal{O}(1/k))}sqrt{2pi (n-k)(1+mathcal{O}(1/(n-k)))}} $



$displaystyle asymp frac{1}{2pi}frac{1}{n}sumlimits_{k=1}^{n-1}frac{1}{sqrt{frac{k}{n}left(1-frac{k}{n}right)}} asymp frac{1}{2pi} intlimits_0^1frac{dx}{sqrt{x(1-x)}} = frac{1}{2}$



Note: I use $asymp$ for "same asymptotically" in terms of $ntoinfty$ .



Maybe someone has an idea, how to simplify that, so that $pi$ and the integral disappear ?!






share|cite|improve this answer























  • How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
    – Diger
    2 hours ago















up vote
0
down vote










up vote
0
down vote









Only a comment.



I also use Stirling and (a Riemann) integral, but in a short way as a rough method.



$displaystyle e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} $



$displaystyle asymp e^{-n}sumlimits_{k=1}^{n-1}frac{e^k e^{n-k}}{sqrt{2pi k (1+mathcal{O}(1/k))}sqrt{2pi (n-k)(1+mathcal{O}(1/(n-k)))}} $



$displaystyle asymp frac{1}{2pi}frac{1}{n}sumlimits_{k=1}^{n-1}frac{1}{sqrt{frac{k}{n}left(1-frac{k}{n}right)}} asymp frac{1}{2pi} intlimits_0^1frac{dx}{sqrt{x(1-x)}} = frac{1}{2}$



Note: I use $asymp$ for "same asymptotically" in terms of $ntoinfty$ .



Maybe someone has an idea, how to simplify that, so that $pi$ and the integral disappear ?!






share|cite|improve this answer














Only a comment.



I also use Stirling and (a Riemann) integral, but in a short way as a rough method.



$displaystyle e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} $



$displaystyle asymp e^{-n}sumlimits_{k=1}^{n-1}frac{e^k e^{n-k}}{sqrt{2pi k (1+mathcal{O}(1/k))}sqrt{2pi (n-k)(1+mathcal{O}(1/(n-k)))}} $



$displaystyle asymp frac{1}{2pi}frac{1}{n}sumlimits_{k=1}^{n-1}frac{1}{sqrt{frac{k}{n}left(1-frac{k}{n}right)}} asymp frac{1}{2pi} intlimits_0^1frac{dx}{sqrt{x(1-x)}} = frac{1}{2}$



Note: I use $asymp$ for "same asymptotically" in terms of $ntoinfty$ .



Maybe someone has an idea, how to simplify that, so that $pi$ and the integral disappear ?!







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 12 hours ago

























answered 12 hours ago









user90369

8,118925




8,118925












  • How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
    – Diger
    2 hours ago




















  • How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
    – Diger
    2 hours ago


















How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
– Diger
2 hours ago






How do you get the equality: $$ e^{-n}sumlimits_{k=0}^nfrac{n^k}{k!} = e^{-n}sumlimits_{k=0}^nfrac{k^k (n-k)^{n-k}}{k!(n-k)!} , ? $$
– Diger
2 hours ago




















 

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