Answer to Question #41536 in Statistics and Probability for anny

Question #41536
hi! please help me out with prob.stuff. got stuck with one problem from my homework, need advice.

We have n letters for constructing words, letter a occurs α times, letter b occurs β times, and the remaining letters are distinct. How many different r-letter words containing h times letter a and k times letter b are possible?
1
Expert's answer
2014-04-18T11:32:44-0400
We may regard the arrangement of letters as a combined outcome of the three separate experiments. Having r positions left in a r-letter word, the first experiment is arranging h times letter a (there are ((h of r))=r!/h!(r-h)! ways to do that). After the first experiment there are (r-h) positions left and we make the second experiment which consists in arranging k times letter b (there are ((k of h-r))=((r-h)!)/k!(r-h-k)! ways to do that). After the second experiment there are (r-h-k) positions left and we arrange distinct remaining n-α-β letters on those positions (there are ((n-α-β)!)/(n-α-β-(r-h-k))! ways to do that). Hence it follows from the basic principle of counting,
r!/h!(r-h)!*((r-h)!)/k!(r-h-k)!*((n-α-β)!)/(n-α-β-(r-h-k))!=(r!(n-α-β)!)/{h!k!(r-h-k)!(n-α-β-(r-h-k))!}=(r!(n-α-β)!)/{h!k!(r-h-k)!(n-α-β-r+h+k)!} different r-letter words containing h times letter a and k times letter b are possible.

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