I am reading the proof of the Plotkin bound on wikipedia which is here . There is a part of the proof which does not seem to clear to me which is as follows:
Let $C$ be a binary $[n,M,d]$ code. Suppose we arrange all the $M$ codewords of $C$ as rows of a matrix $A$ and let $a_{i}$ denote the number of $1$'s in the $i$th column of $A$. Then the number of zeros in the ith column will be $M-a_i$. Let $k$ be the sum of the distances between pairs of distinct codewords in $C$. How does $$k= \sum_{i=1}^{n}a_i(M-a_i) \quad?$$
In column $i$ the pairs of distinct symbols $(0,1),(1,0)$ in distinct rows contribute to the hamming distances. There are eqactly $a_i(M-a_i)$ such pairs.
Now sum over all columns.