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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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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
- Spring '09