How many ways are there to assign $5$ different jobs to $4$ different employees if every employee is assigned at least $1$ job?

We need to assign $5$ different jobs to $4$ different employees such that every employee gets at least one job.

This is an onto function problem.

Total ways without restriction:

$4^5=1024$

Subtract cases where at least one employee gets no job:

$\binom{4}{1}3^5=4\times243=972$

Add cases where two employees get no job:

$\binom{4}{2}2^5=6\times32=192$

Subtract cases where three employees get no job:

$\binom{4}{3}1^5=4$

Required ways:

$4^5-\binom{4}{1}3^5+\binom{4}{2}2^5-\binom{4}{3}1^5$

$=1024-972+192-4=240$