Question

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 $N?$

$9$

$14$

$16$

$17$

$19$

Solution:

A term that includes all four variables has form $a^vb^wc^xd^y$ , with $v,w,x,y\ge1$ , multiplied by some power of $1$ . If that power is $z$ , then $v+w+x+y+z=N.$

After subtracting $1$ from each of $v,w,x,y$ , this becomes a nonnegative stars-and-bars count with $5$ variables summing to $N-4$ . The number of such terms is $\binom{N}{4}.$

Since $\binom{14}{4}=1001$ , we get $N=14$ .

Thus, the correct answer is B.

2016 AMC 10A Problem 20

Solution:

Problem 20 in Other Years