1958 AHSME Problems/Problem 49

Problem

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:

$\textbf{(A)}\ 11\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 55\qquad \textbf{(D)}\ 66\qquad \textbf{(E)}\ 132$

Expand the binomial $((a+b)+c)^n$ with the binomial theorem. We have:

$\sum\limits_{k=0}^{10} \binom{10}{k} (a+b)^k c^{10-k}$

So for each iteration of the summation operator, we add k+1 dissimilar terms. Therefore our answer is:

$\sum\limits_{k=0}^{10} k+1 = \frac{11(1+11)}{2} = 66 \to \boxed{\textbf{D}}$

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 48	Followed by Problem 50
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