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MAT 521Spring Lecture 5

# MAT 521Spring Lecture 5 - Lecture 5 Section 1.9 Multinomial...

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Lecture 5 Section 1.9: Multinomial Coefficients Definition: Any arrangement of n distinct objects into k cells A 1 , A 2 , …, A k such that the first cell A 1 contains n 1 elements, the second cell A 2 contains n 2 elements, …, and the kth cell A k contains n k elements, where n 1 + n 2 + … + n k = n, is called an ordered partitions . The total number of the above ordered partitions is equal to 1 n n - 2 1 n n n - - - - 1 k 2 k 1 n n - ... n n - - k 1 - k 1 n n - ... n n = )! n n ( ! n ! n 1 1 - )! n n n ( ! n )! n n ( 2 1 2 1 - - - )! n ... n n ( ! n )! n ... n n ( 1 k 1 1 k 2 k 1 - - - - - - - - - )! n n ( ! n ! n 1 1 - = ! n !... n ! n ! n k 2 1 = k 2 1 n ... n n n , which is called multinomial coefficient.

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Theorem 1.9.1 Multinomial Theorem For all numbers x 1 , x 2 , …, x k and each positive integer n, (x 1 + x 2 + …+ x k ) n = k 2 1 n k n 2 n 1 k 2 1 x ... x x n ... n n n where the summation extends over all possible combinations of nonnegative integers n 1 , n 2 , …, n k such that n 1 + n 2 +…, + n
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MAT 521Spring Lecture 5 - Lecture 5 Section 1.9 Multinomial...

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