Problem:
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 Choices:
A. 9
B. 14
C. 16
D. 17
E. 19
Solution:
If a term contains all four variables a,b,c, and d, then it has the form ai+1bj+1ck+1dl+11m for some nonnegative integers i,j,k,l, and m such that (i+1)+(j+1)+(k+1)+(l+1)+m=N or i+j+k+l+m=N−4. The number of terms can be counted using the stars and bars technique. The number of linear arrangements of N−4 stars and 4 bars corresponds to the number of possible values of i,j,k,l, and m. Namely, in each arrangement the bars separate the stars into five groups (some of them can be empty) whose sizes are the values of i,j,k,l, and m. There are
(N−4+44)=(N4)=4⋅3⋅2⋅1N(N−1)(N−2)(N−3)=1001=7⋅11⋅13
such arrangements. So N(N−1)(N−2)(N−3)=4⋅3⋅2⋅7⋅11⋅13=14⋅13⋅12⋅11. Thus the answer is N=(B)14.
The problems on this page are the property of the MAA's American Mathematics Competitions