When $(a+b+c+d)^{10}$ is expanded and like terms combined, how many terms are the result?

Question

I came across this problem in my number theory class:

When $(a+b+c+d)^{10}$ is expanded and like terms combined, how many terms are the result?

I don't know what do, or where to start on this problem. Does anybody have any ideas? Thanks.

Answer 1

Every term is of the form $a^ib^jc^kd^l$ where $i+j+k+l=10$ and $0\leq i,j,k,l$.

In general one can solve the following problem for any $n$ and $k$:

How many sequences $0\leq a_1,a_2,\dots a_k$ satisfy $a_1+a_2+\dots + a_k=n$?

This is known as the number of weak compositions of $n$ into $k$ summands.

The number of such compositions is $\binom{n+k-1}{k-1}$.

The method for obtaining this formula is quite beautiful and called Stars and Bars.

So your answer is $\binom{13}{3}=286$