Determine the existence of a relation

Question

I came up with the following relation:

$$ \sum_{i=0}^{\sum_{j=1}^{n} p_j} \binom {\sum_{j=1}^{n} p_j}i = \prod_{i=1}^{n} \sum_{j=0}^{p_i} \binom {p_i}{j} $$

where $p_i$ is an element of a set of $n$ positive integers.

It is probably a known relation yet I am still interested as to the approach one should take to determine the existence of a relation?

Answer 1

We can show equality as follows \begin{align*} \sum_{i=0}^{\sum_{j=1}^{n} p_j} \binom {\sum_{j=1}^{n} p_j}i =2^{\sum_{j=1}^{n} p_j} =\prod_{i=1}^{n} 2^{p_i} =\prod_{i=1}^{n} \sum_{j=0}^{p_i} \binom {p_i}{j} \end{align*}

Here we apply the binomial identity $\sum_{j=0}^n\binom{n}{j}=(1+1)^n=2^n$ twice.