University of South Carolina High School Math Contest/1993 Exam/Problem 8
Problem
What is the coefficient of in the expansion of
![$(1 + x + x^2 + x^3 + x^4 + x^5 )^6?$](http://latex.artofproblemsolving.com/d/9/9/d991d3f18140fdd0a7e63164d02a3ad6edba8731.png)
![$\mathrm{(A) \ } 40 \qquad \mathrm{(B) \ }48 \qquad \mathrm{(C) \ }56 \qquad \mathrm{(D) \ }62 \qquad \mathrm{(E) \ } 64$](http://latex.artofproblemsolving.com/d/3/7/d37b89c36eabc473792f9ed081d97757a8f69f3d.png)
Solution
If we expand out the given product, we see that we have a sum of terms in which each term is a product of six members of the set (with repetitions allowed). In order to have one of these terms equal to
, we can either have a single
term and five terms of 1 in our product (
ways) or one
term, one
term and four 1 terms (
ways) or have three
terms (
ways). This gives us a total of
.