Difference between revisions of "2016 AMC 10A Problems/Problem 20"

Revision as of 22:40, 3 February 2016

For some particular value of $N$ , when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$ , each to some positive power. What is $N$ ?

$\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

Solution

All the desired terms are in the form $a^xb^yc^zd^w1^t$ , where $x + y + z + w + t = N$ (the $1^t$ part is necessary to make stars and bars work better.) Since $x$ , $y$ , $z$ , and $w$ must be at least $1$ , let $x' = x - 1$ , $y' = y - 1$ , $z' = z - 1$ , and $w' = w - 1$ , so $x' + y' + z' + w' + t = N - 4$ . Now, we use stars and bars to see that there are $\binom{N}{4}$ solutions to this equation. We have $\binom{14}{4} = 1001$ , so $N = \boxed{14}$ .

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 ==See Also==
-{{AMC10 box|year=2016|ab=A|num-b=23|num-a=25}}
+{{AMC10 box|year=2016|ab=A|num-b=19|num-a=21}}
 {{MAA Notice}}

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Difference between revisions of "2016 AMC 10A Problems/Problem 20"

Revision as of 22:40, 3 February 2016

Solution

See Also