2008 AMC 12A Problems/Problem 19

Problem

In the expansion of

$\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2$ ,

what is the coefficient of $x^{28}$ ?

$\textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405$

Solution

Let $A = \left(1 + x + x^2 + \cdots + x^{14}\right)$ and $B = \left(1 + x + x^2 + \cdots + x^{27}\right)$ . We are expanding $A \cdot A \cdot B$ .

Since there are $15$ terms in $A$ , there are $15^2 = 225$ ways to choose one term from each $A$ . The product of the selected terms is $x^n$ for some integer $n$ between $0$ and $28$ inclusive. For each $n \neq 0$ , there is one and only one $x^{28 - n}$ in $B$ . Since there is only one way to choose one term from each $A$ to get a product of $x^0$ , there are $225 - 1 = 224$ ways to choose one term from each $A$ and one term from $B$ to get a product of $x^{28}$ . Thus the coefficient of the $x^{28}$ term is $224 \Rightarrow C$ .

2008 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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2008 AMC 12A Problems/Problem 19

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