In the expansion of (1 + x + x^2 + \cdots + x^27)(1 + x + x^2 + \cdots + x^14)^2, what is the coefficient of x^28?

2008 AMC 12A Problem 19 — Solution

2008 AMC 12A Problem 19

Below is the professionally curated solution for Problem 19 of the 2008 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12A solutions, or check the answer key.

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Concepts:generating functions partitions and compositions

Difficulty rating: 1930

19.

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 $x^{28}?$

$195$

$196$

$224$

$378$

$405$

Solution:

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.

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