2016
Problem - 2884
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 $N$?
Answer
14
After expansion, all the merged terms must be in the form of $a^{u}b^{v}c^{w}d^{x}1^{y}$ where integers $u + v + w + x + y = N$.
To satisfy the requirement, $u$, $v$, $w$, $x$ must be positive and $y$ can be either $0$ or positive. Let $z=y+1$, then the number of terms equals to the positive integer solutions to the following equation: $$u + v + w + x + z = N +1$$
This is a basic pattern and the answer is $\myBinom{N}{4}$. Hence, $$\myBinom{N}{4} = 1001\implies N=\boxed{14}$$