The count of good “words”
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0
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How many good $words$ are there , which have length $n$ and consist of {0,1}
A $word$ is considered good if the number of occurrences of $0$ in the $word$ is $even$:
ex.
$${
n = 3: [111],[100],[010],[001]
}$$
I have no idea what to do.
Any help would be appreciated
combinatorics combinations
asked Nov 18 at 13:49
R0xx0rZzz
516
add a comment |
up vote
0
down vote
favorite
How many good $words$ are there , which have length $n$ and consist of {0,1}
A $word$ is considered good if the number of occurrences of $0$ in the $word$ is $even$:
ex.
$${
n = 3: [111],[100],[010],[001]
}$$
I have no idea what to do.
Any help would be appreciated
combinatorics combinations
asked Nov 18 at 13:49
R0xx0rZzz
516
Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How many good $words$ are there , which have length $n$ and consist of {0,1}
A $word$ is considered good if the number of occurrences of $0$ in the $word$ is $even$:
ex.
$${
n = 3: [111],[100],[010],[001]
}$$
I have no idea what to do.
Any help would be appreciated
combinatorics combinations
asked Nov 18 at 13:49
R0xx0rZzz
516
How many good $words$ are there , which have length $n$ and consist of {0,1}
A $word$ is considered good if the number of occurrences of $0$ in the $word$ is $even$:
ex.
$${
n = 3: [111],[100],[010],[001]
}$$
I have no idea what to do.
Any help would be appreciated
combinatorics combinations
combinatorics combinations
asked Nov 18 at 13:49
R0xx0rZzz
516
asked Nov 18 at 13:49
R0xx0rZzz
516
asked Nov 18 at 13:49
R0xx0rZzz
516
asked Nov 18 at 13:49
R0xx0rZzz
516
asked Nov 18 at 13:49
R0xx0rZzz
516
516
Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07
add a comment |
Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07
Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07
Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07
add a comment |
1 Answer
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You need all the possible combinations that have 0, 2, 4, ... up to n digits of zero.
Number of possible numbers with 0 digits is $binom{n}{0}$, number of possible numbers with 2 digits $binom{n}{2}$, and so on.
After you sum up all these numbers, you get all the possible numbers that have from 0 to n even digits of 0.
$$sum_{i=0}^frac{n}{2}binom{n}{2i}$$
answered Nov 18 at 20:10
Erik Cristian Seulean
456
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You need all the possible combinations that have 0, 2, 4, ... up to n digits of zero.
Number of possible numbers with 0 digits is $binom{n}{0}$, number of possible numbers with 2 digits $binom{n}{2}$, and so on.
After you sum up all these numbers, you get all the possible numbers that have from 0 to n even digits of 0.
$$sum_{i=0}^frac{n}{2}binom{n}{2i}$$
answered Nov 18 at 20:10
Erik Cristian Seulean
456
add a comment |
up vote
1
down vote
accepted
You need all the possible combinations that have 0, 2, 4, ... up to n digits of zero.
Number of possible numbers with 0 digits is $binom{n}{0}$, number of possible numbers with 2 digits $binom{n}{2}$, and so on.
After you sum up all these numbers, you get all the possible numbers that have from 0 to n even digits of 0.
$$sum_{i=0}^frac{n}{2}binom{n}{2i}$$
answered Nov 18 at 20:10
Erik Cristian Seulean
456
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You need all the possible combinations that have 0, 2, 4, ... up to n digits of zero.
Number of possible numbers with 0 digits is $binom{n}{0}$, number of possible numbers with 2 digits $binom{n}{2}$, and so on.
After you sum up all these numbers, you get all the possible numbers that have from 0 to n even digits of 0.
$$sum_{i=0}^frac{n}{2}binom{n}{2i}$$
answered Nov 18 at 20:10
Erik Cristian Seulean
456
You need all the possible combinations that have 0, 2, 4, ... up to n digits of zero.
Number of possible numbers with 0 digits is $binom{n}{0}$, number of possible numbers with 2 digits $binom{n}{2}$, and so on.
After you sum up all these numbers, you get all the possible numbers that have from 0 to n even digits of 0.
$$sum_{i=0}^frac{n}{2}binom{n}{2i}$$
answered Nov 18 at 20:10
Erik Cristian Seulean
456
answered Nov 18 at 20:10
Erik Cristian Seulean
456
answered Nov 18 at 20:10
Erik Cristian Seulean
456
answered Nov 18 at 20:10
Erik Cristian Seulean
456
456
add a comment |
add a comment |
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Hint: Arrange the words of length n in pairs, so that the words in each pair are the same except for the first digit.
– Michael Behrend
Nov 18 at 15:07