2016 AMC 10A Problem 20

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

Count exponent choices for terms containing all four variables

Stars and bars gives

\binom N4=1001

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.

Problem 20 in other years