Bit Counting Machine

A bit counting machine reads a binary input vector (a, b, c, d, e) and produces a binary number (s₂, s₁, s₀) that counts the number of 1s in the input bits.

For example when (a, b, c, d, e) = (0, 1, 1, 1, 1), we have output (s₂, s₁, s₀) = (1, 0, 0), and when (a, b, c, d, e) = (1, 1, 0, 1, 0), we have output (s₂, s₁, s₀) = (0, 1, 1).

Derive the canonical sum of minterms expression of the three output bits.