JEE Main 2026 — 4 April, Morning Shift — Question 27

Total functions from a set with 4 elements to a set with 3 elements: $3^4 = 81$. Number of onto functions: $3! \cdot S(4,3) = 6 \times 6 = 36$, where $S(4,3)$ is Stirling number of the second kind. Alternatively, number of onto functions = $3^4 - \binom{3}{1}2^4 + \binom{3}{2}1^4 = 81 - 3 \times 16 + 3 \times 1 = 81 - 48 + 3 = 36$. Number of functions not onto = Total functions - onto functions = $81 - 36 = 45$.