How many onto functions $f : \{1, 2, 3, 4\} \to \{p, q\}$ have $f(1) = p$?

Explanation

1. 01

   With $f(1) = p$, each of 2, 3 and 4 goes to $p$ or $q$: $2^{3} = 8$ ways.

2. 02

   Onto needs $q$ to be reached, so drop the one way that sends all three to $p$: $8 - 1 = 7$.