1958 AMC 12 Problem 49

Question

In the expansion of

(a+b)^n

there are

n+1

dissimilar terms. The number of dissimilar terms in the expansion of

(a+b+c)^{10}

is:

11

33

55

66

132

A term is determined by nonnegative exponents

i,j,k

whose sum is

10

Count the solutions of

i+j+k=10

by stars and bars

Each distinct term is

a^ib^jc^k

for nonnegative integers satisfying

i+j+k=10.

By stars and bars, the number of such triples is

\binom{10+3-1}{3-1}=\binom{12}{2}=66.

Thus, the correct answer is D.

Problem 49 in other years