{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Solutions from a student (dragged) 26

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

This preview shows page 1. Sign up to view the full content.

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 coe ffi cient) To compute n n 1 , n 2 , n 3 , · · · , n r we consider fixing one particular object from the n . Then this object can end up in any of the r individual groups. If it appears in the first one then we have n - 1 n 1 - 1 , n 2 , n 3 , · · · , n r , possible arrangements for the other objects. If it appears in the second group then the remaining objects can be distributed in n - 1 n 1 , n 2 - 1 , n 3 , · · · , n r ways, etc. Repeating this argument for all of the r groups we see that the original multinomial coe ffi cient can be written as sums of these individual multinomial terms as n n 1 , n 2 , n 3 , · · · , n r = n - 1 n 1 - 1 , n 2 , n 3 , · · · , n r + n - 1 n 1 , n 2 - 1 , n 3 , · · · , n r + · · · + n - 1 n 1 , n 2 , n 3 , · · · , n r - 1 . Problem 19 (the multinomial theorem) The multinomial therm is ( x 1 + x 2 + · · · + x r ) n = n 1 + n 2 + ··· + n r = n n n 1 , n 2 , · · · , n r x n 1 1 x n 2 2 · · · x n r r , which can be proved by recognizing that the product of (
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online