given two sets how many ways can we choose 2 subsets of same length?

Suppose $|A| = n, |B| = m$ (and that $n \leq m$). We want to pick a two subsets of size $k$, one from $A$ and one from $B$. Note that we can do this in $\binom{n}{k} \binom{m}{k}$ ways; summing over all $k$ gives $\sum\limits_{k=0}^n \binom{n}{k} \binom{m}{k}$.

But it turns out that there is a closed-form expression for this. Note that $\binom{n}{k} = \binom{n}{n-k}$. So our sum is equal to $\sum\limits_{k=0}^n \binom{n}{n-k} \binom{m}{k}$. In other words, how many ways can we choose $k$ elements of $B$ and $n-k$ elements of $A$ as $k$ ranges over all possible values? But this merely asks how many ways are there to pick $n$ elements from $A \cup B$ (assuming the sets are disjoint), which is $\binom{m+n}{n}$. Intuitively, this counts the number of ways to choose $n-k$ elements of $A$ to exclude and $k$ elements of $B$ to include.