2022/23 AHSMC Part 1 Problem 9

Let $p$ be the number of pigs and $d$ be the number of ducks, so $4p + 2d = 40$ or $2p + d = 20$. From this scenario, there are $11$ different combinations for $p$ and $d$ which are $(0,20)$, $(1,18)$, $(2,16)$, $(3,14)$, $(4,12)$, $(5,10)$, $(6,8)$, $(7,6)$, $(8,4)$, $(9,2)$, and $(10,0)$ so we conclude that there are $11$ barns at the farm.

Therefore, there are $4$ $\bigl((1,18)$, $(3,14)$, $(7,6)$, and $(9,2)\bigr)$ out of $11$ barns such that the total number of pigs and ducks is a prime number. The answer is (B).