Unformatted text preview: In general, to divide n objects into one group of n 1 , one group of n 2 , ... , and a k th group of n k , where n = n 1 + ··· + n k , the answer is n ! n 1 ! n 2 ! ··· n k ! . These are known as multinomial coefficients. Suppose one has 8 indistinguishable balls. How many ways can one put them in 3 boxes? Let us make sequences of o ’s and  ’s; any such sequence that has  at each side, 2 other  ’s, and 8 o ’s represents a way of arranging balls into boxes. For example, if one has  o o  o o o  o o o  , this would represent 2 balls in the first box, 3 in the second, and 3 in the third. Altogether there are 8 + 4 symbols, the first is a  as is the last. so there are 10 symbols that can be either  or o . Also, 8 of them must be o . How many ways out of 10 spaces can one pick 8 of them into which to put a o ? The answer is 10 8 . Example. How many quintuples ( x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) of nonnegative integers whose sum is 20 are there?...
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 Spring '09
 wen
 NK, ﬁrst box

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