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
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

Page

Toolbox

Search

2008 AMC 12A Problems/Problem 19

Problem

Solution

See Also