How to get value of this binomial coefficient expression?

Question

I am trying to work out an upper bound (big O) of an algorithm I thought of in graph theory field. Basically I have a graph $G=(V,E)$. And a subset of vertices $A=\{a_1,a_2,...,a_k\} ∈ V$ such that $deg(a_i) > 2$ and $\sum^{k}_{i=1} deg(a_i) ≤ 2|E|$. ($deg(a)$ is the degree of vertex $a$)

So then what would be the exact value for this (like write it without the binomial coefficient)?

$\prod^{k}_{i=1} {deg(a_i) \choose 2}$

This is the number of steps needed for the algorithm to run.

Thanks

Answer 1

Using $\binom{n}{2}=n(n-1)/2$ you get $\frac{1}{2^k}\prod_{i=1}^k (\deg a_i)(\deg a_i - 1)$.

If $\deg a_i>2$ for $1\le i\le k$, then the value of the expression is at least $3^k$.