Say we need to find an expression for the number of binary strings of length $n$, which have at most $k$ ones. My solution was to split the problem into $k+1$ cases, where the number of ones, $m=0,1,2,...,k$, and sum the values together.
For example, to find the number of solutions in which at most $3$ ones occur in a $6$ bit string, we compute $\binom{6}{0}+\binom{6}{1}+\binom{6}{2}+\binom{6}{3}=1+6+15+20=42$. More generally,
$$ N=\sum_{m=0}^{k}\binom{n}{m} $$
This is pretty nice, but I'd like to find a nicer expression which doesn't involve a summation, if such a thing exists. Any suggestions?