In the expansion of &(1 + x + x^2 + … + x^27) & × (1 + x + x^2 + … + x^14)^2, what is the coefficient of x^28?

2008 AMC 12A Problem 19

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19.

In the expansion of

\begin{aligned} &\left(1 + x + x^2 + \cdots + x^{27}\right) \\ &\quad {}\cdot \left(1 + x + x^2 + \cdots + x^{14}\right)^2, \end{aligned}

what is the coefficient of

x^{28}?

195

196

224

378

405

A term has the form

x^{a+b+c}

with

0 \le a \le 27

and

0 \le b, c \le 14

For each of the

15^2

pairs

(b, c),

there is a unique valid

a,

except when

b = c = 0

Each term is

x^{a + b + c}

with

0 \le a \le 27

and

0 \le b, c \le 14.

To get

x^{28}

we need

a = 28 - b - c.

There are

(14 + 1)^2 = 225

choices for

(b, c).

For every choice except

(b, c) = (0, 0),

the required

a = 28 - b - c

lies in

[0, 27],

giving a valid term.

The coefficient of

x^{28}

is therefore

225 - 1 = 224.

Thus, C is the correct answer.