JEE 2023 — Mathematics PYQ

Total number of onto functions

$$=\frac{5!}{3!2!}\times 4!$$

$$=\frac{5\times 4}{2}\times 24=240$$

When $f(a)=1$, number of onto functions

$=4!+\frac{4!}{2!2!}\times 3!$

=24+36=60

So, required number of onto functions

=240-60=180