Mixing
Finding the number of combinations that fit certain criteria
$begingroup$
Assuming that the letters A, B and C are used to generate all possible "strings" of length seventeen (17) characters (like ACBBCABCACBCABCCB), how many of these strings have exactly four (4) B's? How can we go about this?
Context: I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
combinatorics permutations
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
$endgroup$
add a comment |
$begingroup$
Assuming that the letters A, B and C are used to generate all possible "strings" of length seventeen (17) characters (like ACBBCABCACBCABCCB), how many of these strings have exactly four (4) B's? How can we go about this?
Context: I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
combinatorics permutations
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
$endgroup$
$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
2
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03
add a comment |
$begingroup$
Assuming that the letters A, B and C are used to generate all possible "strings" of length seventeen (17) characters (like ACBBCABCACBCABCCB), how many of these strings have exactly four (4) B's? How can we go about this?
Context: I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
combinatorics permutations
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
$endgroup$
Assuming that the letters A, B and C are used to generate all possible "strings" of length seventeen (17) characters (like ACBBCABCACBCABCCB), how many of these strings have exactly four (4) B's? How can we go about this?
Context: I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
combinatorics permutations
combinatorics permutations
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
edited Jan 1 at 21:19
N. F. Taussig
43.7k93355
43.7k93355
asked Jan 1 at 19:05
Muga S.Muga S.
1
asked Jan 1 at 19:05
Muga S.Muga S.
1
asked Jan 1 at 19:05
Muga S.Muga S.
1
1
$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
2
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03
add a comment |
$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
2
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03
$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
2
2
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03
add a comment |
1 Answer
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$begingroup$
First pick the places for the $B$'s, which you can do in ${17 choose 4}=2380$ ways. Then each of the other places has two choices, so that gives a factor $2^{13}=8192$. Multiply them.
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
$endgroup$
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
add a comment |
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1 Answer
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$begingroup$
First pick the places for the $B$'s, which you can do in ${17 choose 4}=2380$ ways. Then each of the other places has two choices, so that gives a factor $2^{13}=8192$. Multiply them.
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
$endgroup$
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
add a comment |
$begingroup$
First pick the places for the $B$'s, which you can do in ${17 choose 4}=2380$ ways. Then each of the other places has two choices, so that gives a factor $2^{13}=8192$. Multiply them.
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
$endgroup$
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
add a comment |
$begingroup$
First pick the places for the $B$'s, which you can do in ${17 choose 4}=2380$ ways. Then each of the other places has two choices, so that gives a factor $2^{13}=8192$. Multiply them.
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
$endgroup$
First pick the places for the $B$'s, which you can do in ${17 choose 4}=2380$ ways. Then each of the other places has two choices, so that gives a factor $2^{13}=8192$. Multiply them.
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
answered Jan 1 at 23:17
Ross MillikanRoss Millikan
293k23197371
293k23197371
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
add a comment |
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
$begingroup$
Thank you. That works. Appreciated.
$endgroup$
– Muga S.
Jan 2 at 0:27
add a comment |
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$begingroup$
Is this a homework problem? Further context would be appreciated.
$endgroup$
– Zachary Hunter
Jan 1 at 19:09
$begingroup$
No, not a homework problem. I recently came across a piece of software that generates all possible combinations from a given character set. I used it to generate all the possible strings that are 17-character long with the character set A, B, C. A total of over 129 million combinations were generated, the resulting text file being over 2GB! It would be infeasible to go through all of those combinations by hand to find all the strings that have 4 B's. As such, I am looking for a way to solve that problem mathematically.
$endgroup$
– Muga S.
Jan 1 at 20:24
2
$begingroup$
There are $3^{17}$ strings of length $17$ that can be formed using the alphabet ${A, B, C}$ since there are three choices for each position. In your problem, you must choose four positions for the $B$s. You are then left with two choices for each of the remaining $13$ positions.
$endgroup$
– N. F. Taussig
Jan 1 at 21:21
$begingroup$
So, going by your theory, there are 2^13 strings with 4 B's, am I right?
$endgroup$
– Muga S.
Jan 1 at 21:23
$begingroup$
Instead of generating strings of length 17 characters, I toned it down to 8 characters just to have a fewer number of possible combinations to work with. Still using the character set A, B, C, that's a total of 3^8 = 6,561 total combinations. How many of these combinations have 4 B's? Well, 2^4, right? Turns out, no. I generated all those 6,561 strings, pasted them to Microsoft Excel and asked it to count the number of B's in all the strings. Then I just copied the ones with 4 B's to a new Excel worksheet. I get 1,120 strings with 4 B's but not 2^4. What gives?
$endgroup$
– Muga S.
Jan 1 at 22:03