Let $H$ be a finite group with an identity element. Suppose that $\#H = n$ and $(a_1,\ldots,a_n)\in H^n= H\times\cdots \times H$ ($n$ times). It is to prove there exists $p$,$q$ with $0 \leq p<q\leq n$ such that the products $a_1a_2\cdots a_p$ $=$ $a_1a_2\cdots a_p\cdots a_q$.
How to apply Pigeonhole problem to it? Will the element and their respective inverses form a pair and hence a hole eg. $(a_1,a_1^{-1}),(a_2,a_2^{-1})\cdots(a_r,a_p^{-1})$ and try to fit $n$ pigeons that is $(a_1,\ldots ,a_n)$ into those holes?
As in Nex's hint, let $x_i = a_1a_2\dots a_i$, with $0 \leq i \leq n$ (where $x_0$ is simply the identity element). Since the index $i$ can assume $n+1$ values and $H$ has order $n$, for at least one pair of distinct indices $0 \leq p < q \leq n$ it must be that $x_p = x_q$ (for otherwise, one would have $n+1$ distinct elements in $H$).