Given a triangle $K \subset \mathbb{R}^2$ with vertices $A,B,C$ let $\phi_A, \phi_B, \phi_C: K \to \mathbb{R}$ be hat basis functions. They are affine functions defined at nodes such that, for example: $\phi_A(A)=1$ and $\phi_A(B)=\phi_A(C)=0$, followed by linear interpolation.
In other words, they each form a "hat" above the triangle. Here's a link to a visualisation of hat basis functions. It comes from a related article, asking about a special case of this question.
The problem is to prove: $$\int_K\phi_A^m\phi_B^n\phi_C^pdxdy=\frac{2m!n!p!}{(m+n+p+2)!}|K|$$
It is an equation 3.64 from the book The Finite Element Method: Theory, Implementation, and Applications . It is provided without proof, with only a claim that it should be easy to prove using induction.