Combinations of X's and Y'sDate: 10/27/1999 at 09:58:42 From: John Dodgson Subject: GCSE coursework (arrangements and formula) A number of X's and a number of Y's are written in a row such as XX...XXYY...Y. Investigate the number of different arrangements of the letters and find a formula for it. I tried writing out the different arrangements, but I'm not sure of the arrangements or the formula for the number of the arrangements. Here are some of the formulas I have come up with: C = n^2 C = x!/y! and C = n! / [(number of X's)! * (number of Y's)!] Please can you help me? Thanks Date: 10/27/1999 at 11:42:53 From: Doctor Anthony Subject: Re: GCSE coursework (arrangements and formula) Let's begin with a general note on permutations and combinations from our archives: Permutations and Combinations: Deriving Formulae http://mathforum.org/dr.math/problems/lim6.18.98.html So if you have m X's and n Y's, in how many DIFFERENT ways could these m+n letters be arranged? In the case of m X's and n Y's, if all the X's were different and all the Y's were different there would be m+n different letters to permute, and this could be done in (m+n)! ways. However, since the X's are all the same, they could be permuted amongst themselves in m! ways and the answer (m+n)! is too large by a factor of m!. Similarly, the Y's are all the same and could be permuted amongst themselves in n! ways without giving rise to a new arrangement, and (m+n)! is too large by a further factor of n!. So the actual number of different arrangements is: (m+n)! -------- m! n! Another example: in how many different ways can the letters of the word STEEPLES be arranged? There 8 letters. 3 E's, 2 S's and 3 other letters. 8! Number of arrangements = ----- = 3360 3! 2! - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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