Solutions from a student (dragged) 26

Solutions from a student (dragged) 26 - , which can be...

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objects into two groups of size r and of size n - r respectively. As these are the same thing the numbers are equivalent. Problem 18 (a decomposition of the multinomial coeFcient) To compute ± n n 1 ,n 2 ,n 3 , ··· ,n r ² we consider Fxing one particular object from the n .Then this object can end up in any of the r individual groups. If it appears in the Frst one then we have ± n - 1 n 1 - 1 ,n 2 ,n 3 , ··· ,n r ² ,poss ib learrangementsfortheotherobjects .Ifitappears in the second group then the remaining objects can be distributed in ± n - 1 n 1 ,n 2 - 1 ,n 3 , ··· ,n r ² ways, etc. Repeating this argument for all of the r groups we see that the original multinomial coe±cient can be written as sums of these individual multinomial terms as ± n n 1 ,n 2 ,n 3 , ··· ,n r ² = ± n - 1 n 1 - 1 ,n 2 ,n 3 , ··· ,n r ² + ± n - 1 n 1 ,n 2 - 1 ,n 3 , ··· ,n r ² + ··· + ± n - 1 n 1 ,n 2 ,n 3 , ··· ,n r - 1 ² . Problem 19 (the multinomial theorem) The multinomial therm is ( x 1 + x 2 + ··· + x r ) n = ³ n 1 + n 2 + ··· + n r = n ± n n 1 ,n 2 , ··· ,n r ² x n 1 1 x n 2 2 ··· x
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Unformatted text preview: , which can be proved by recognizing that the product of ( x 1 + x 2 + + x r ) n will contain products of the type x n 1 1 x n 2 2 x n r r , and recognizing that the number of such terms, i.e. the coecient in front of this term is a count of the number of times we can select n 1 of the variable x 1 s, and n 2 of the variable x 2 , etc from the n variable choices. Since this number equals the multinomial coecient we have proven the multinomial theorem. Problem 20 (the number of ways to ll bounded urns) Let x i be the number of balls in the i th urn. We must have x i m i and we are distributing the n balls so that r i =1 x i = n . To solve this problem lets shift our variables so that each must be greater than or equal to zero. Our constraint then becomes (by subtracting the...
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This note was uploaded on 02/25/2011 for the course STAT 418 taught by Professor G.jogeshbabu during the Winter '08 term at Pennsylvania State University, University Park.

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