Question

The number of surjective functions from $A$ to $B$ where $A = \\{1, 2, 3, 4 \\}$ and $B = \\{a, b\\}$ is

• A. 14
• B. 12
• C. 2
• D. 15

Answer 1

Answer: (A)

14

Answer 2

If A and B are two sets having m and n elements such that
$1 \le n \le m = \sum^{n}_{r=1} (-1)^{n-r} {^nC_{r}} r^{m}$
$= \sum^{n}_{r=1}(-1)^{2-r} {^{2}C_{r}} (r)^{4}$
$= (-1)^{2-1} {^{2}C_{1}} (1)^{4} + (-1)^{2-2} {^{2}C_{2}} (2)^{4} = -2 + 16$
$= 14$