Difference between revisions of "2008 AMC 12A Problems/Problem 19"

Revision as of 22:54, 30 December 2015

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 $x^{28}$ ?

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

Solution 1

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 $B$ . For example, if I choose $x^2$ from $A$ , then there is exactly one power of $x$ in $B$ that I can choose; in this case, it would be $x^{24}$ . 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 $224 \Rightarrow C$ .

Solution 2

Let $P(x) = \left(1 + x + x^2 + \cdots + x^{14}\right)^2 = a_0 + a_1x + a_2x^2 + \cdots + a_{28}x^{28}$ . Then the $x^{28}$ term from the product in question $\left(1 + x + x^2 + \cdots + x^{27}\right)(a_0 + a_1x + a_2x^2 + \cdots + a_{28}x^{28})$ is

$1a_{28}x^{28} + xa_{27}x^{27} + x^2a_{26}x^{26} + \cdots + x^{27}a_1x = \left(a_1 + a_2 + \cdots a_{28}\right)x^{28}$

So we are trying to find the sum of the coefficients of $P(x)$ minus $a_0$ . Since the constant term $a_0$ in $P(x)$ (when expanded) is $1$ , and the sum of the coefficients of $P(x)$ is $P(1)$ , we find the answer to be $P(1) - a_0 = \left(1 + 1 + 1^2 + \cdots 1^{14}\right)^2 - 1 = 15^2 - 1 = 224 \rightarrow C$

Solution 3

We expand $(1 + x + x^2 + x^3 + \cdots + x^{14})^2$ to $(1 + x + x^2 + x^3 + \cdots + x^{14}) * (1 + x + x^2 + x^3 + \cdots + x^{14})$ and use FOIL to multiply. It expands out to:

$1 + x + x^2 + x^3 + x^4 + \cdots + x^{14} +$

$\qquad x + x^2 + x^3 + x^4 + \cdots + x^{14} + x^{15} +$

$\qquad \qquad x^2 + x^3 + x^4 + \cdots + x^{14} + x^{15} + x^{16} + \cdots$

It becomes apparent that

$(1 + x + x^2 + x^3 + \cdots + x^{14})^2 = 1 + 2x + 3x^2 + 4x^3 + \cdots + 15x^{14} + 14x^{15} + 13x^{16} + \cdots + x^{28}$ .

Now we have to find the coefficient of $x^{28}$ in the product:

$(1 + 2x + 3x^2 + 4x^3 + \cdots + 15x^{14} + 14x^{15} + 13x^{16} + \cdots + x^{28}) * (1 + x + x^2 + x^3 + \cdots + x^{27})$ .

We quickly see that the we get $x^{28}$ terms from $x^{27} * 2x$ , $x^{26} * 3x^2$ , $x^{25} * 4x^3$ , ... $15x^{14} * x^{14}$ , ... $x^{28} * 1$ . The coefficient of $x^{28}$ is just the sum of the coefficients of all these terms. $1 + 2 + 3 + 4 + \cdots + 15 + 14 + 13 + \cdots + 4 + 3 + 2 = 224$ , so the answer is $\boxed{C}$ .

Solution 4

Rewrite the product as $\frac{(x^{28} - 1)(x^{15} - 1)(x^{15} - 1)}{(x - 1)^3}$ . It is known that

$\frac{1}{(1 - x)^n} = \binom{n - 1}{n - 1} + \binom{n}{n - 1}x + \binom{n + 1}{n - 1}x^2 + \binom{n + 2}{n - 1}x^3 + \cdots + \binom{n - 1 + k}{n - 1}x^k + \cdots .$

Thus, our product becomes

$-\left( x^{28} - 1 \right) \left( x^{15} - 1 \right) \left( x^{15} - 1 \right) \left( \binom{2}{2} + \binom{3}{2}x + \binom{4}{2}x^2 + \cdots \right).$

$= -\left( x^{28} - 1 \right) \left( x^{15} - 1 \right) \left( x^{15} - 1 \right) \left( 1 + 3x + 6x^2 + \cdots \right).$

We determine the $x^{28}$ coefficient by doing casework on the first three terms in our product. We can obtain an $x^{28}$ term by choosing $x^{28}$ in the first term, $-1$ in the second and third terms, and $1$ in the fourth term. We can get two $x^{28}$ terms by choosing $x^{15}$ in either the second or third term, $-1$ in the first term, $-1$ in the second or third term from which $x^{15}$ has not been chosen, and the $\binom{15}{2}x^{13}$ in the fourth term. We get $\binom{15}{2} * 2 = 210$ $x^{28}$ terms this way. (We multiply by $2$ because the $x^{15}$ term could have been chosen from the second term or the third term). Lastly, we can get an $x^{28}$ term by choosing $-1$ in the first three terms and a $\binom{30}{2}x^{28}$ from the fourth term. We have a total of $1 + 210 - 435 = -224$ for the $x^{28}$ coefficient, but we recall that we have a negative sign in front of our product, so we obtain an answer of $224$ $\boxed{C)}$ .

@@ Line 40: / Line 40: @@
 We quickly see that the we get <math>x^{28}</math> terms from <math>x^{27} * 2x</math>, <math>x^{26} * 3x^2</math>, <math>x^{25} * 4x^3</math>, ... <math>15x^{14} * x^{14}</math>, ... <math>x^{28} * 1</math>. The coefficient of <math>x^{28}</math> is just the sum of the coefficients of all these terms. <math>1 + 2 + 3 + 4 + \cdots + 15 + 14 + 13 + \cdots + 4 + 3 + 2 = 224</math>, so the answer is <math>\boxed{C}</math>.
+==Solution 4==
+Rewrite the product as <math>\frac{(x^{28} - 1)(x^{15} - 1)(x^{15} - 1)}{(x - 1)^3}</math>. It is known that
+<cmath>\frac{1}{(1 - x)^n} = \binom{n - 1}{n - 1} + \binom{n}{n - 1}x + \binom{n + 1}{n - 1}x^2 + \binom{n + 2}{n - 1}x^3 + \cdots + \binom{n - 1 + k}{n - 1}x^k + \cdots .</cmath>
+Thus, our product becomes
+<cmath>-\left( x^{28} - 1 \right) \left( x^{15} - 1 \right) \left( x^{15} - 1 \right) \left( \binom{2}{2} + \binom{3}{2}x + \binom{4}{2}x^2 + \cdots \right).</cmath>
+<cmath>= -\left( x^{28} - 1 \right) \left( x^{15} - 1 \right) \left( x^{15} - 1 \right) \left( 1 + 3x + 6x^2 + \cdots \right). </cmath>
+We determine the <math>x^{28}</math> coefficient by doing casework on the first three terms in our product. We can obtain an <math>x^{28}</math> term by choosing <math>x^{28}</math> in the first term, <math>-1</math> in the second and third terms, and <math>1</math> in the fourth term. We can get two <math>x^{28}</math> terms by choosing <math>x^{15}</math> in either the second or third term, <math>-1</math> in the first term, <math>-1</math> in the second or third term from which <math>x^{15}</math> has not been chosen, and the <math>\binom{15}{2}x^{13}</math> in the fourth term. We get <math>\binom{15}{2} * 2 = 210</math> <math>x^{28}</math> terms this way. (We multiply by <math>2</math> because the <math>x^{15}</math> term could have been chosen from the second term or the third term). Lastly, we can get an <math>x^{28}</math> term by choosing <math>-1</math> in the first three terms and a <math>\binom{30}{2}x^{28}</math> from the fourth term. We have a total of <math>1 + 210 - 435 = -224</math> for the <math>x^{28}</math> coefficient, but we recall that we have a negative sign in front of our product, so we obtain an answer of <math>224</math> <math>\boxed{C)}</math>.
 ==See Also==

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

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