I believe that I saw this problem not long ago in a book:
Solve the Diophantine Equation $k!=n(n+1)(n+2)$, where $k,n$ are positive integers.
However, I am no longer able to find this question, and further examination has revealed the possibility that I may have been mistaken.
The equation appears to be similar to Brocard`s Problem, a unsolved problem in mathematics.
The only solutions appear to be $(n,k)=(1,3)(8,6)(4,5)(2,4)$.
Is there a easy way to solve this problem?