Unformatted text preview: , which can be proved by recognizing that the product of ( x 1 + x 2 + Â· Â· Â· + x r ) n will contain products of the type x n 1 1 x n 2 2 Â· Â· Â· x n r r , and recognizing that the number of such terms, i.e. the coeÂ±cient in front of this term is a count of the number of times we can select n 1 of the variable x 1 â€™s, and n 2 of the variable x 2 , etc from the n variable choices. Since this number equals the multinomial coeÂ±cient we have proven the multinomial theorem. Problem 20 (the number of ways to Â±ll bounded urns) Let x i be the number of balls in the i th urn. We must have x i â‰¥ m i and we are distributing the n balls so that âˆ‘ 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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 Winter '08
 G.JOGESHBABU
 Algebra, Probability, Equals sign, South African National Roads Agency, original multinomial coeï¬ƒcient, individual multinomial terms

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