How many ways are there to assign 5 different jobs to 4 different employees if…

Answer: 240 ways.

Method 1 (Inclusion–Exclusion):

• Total assignments without restrictions: 4^5 = 1024.

• Use inclusion–exclusion to remove assignments that leave one or more employees with zero jobs.

• 4^5 - C(4,1)·3^5 + C(4,2)·2^5 - C(4,3)·1^5 = 1024 - 4·243 + 6·32 - 4·1 = 240.

Method 2 (Stirling numbers):

• Number of ways to partition 5 distinct jobs into 4 nonempty unlabeled groups is S(5,4) = 10.

• Label the 4 groups by the 4 distinct employees: multiply by 4! = 24, giving 10·24 = 240.

Both methods agree, so the number of ways is 240.