In a task assignment situation, in how many different ways can five jobs be assigned to five machines if each job must be assigned to each machine and each machine must be assigned one job?

120

The simple answer is that 5 items can be arranged 5! (5 factorial) different ways. But let's expand upon that brief answer. We have 5 jobs and 5 machines with which to perform those jobs. So let's look at the 1st machine. Any of 5 of the jobs may be assigned to it. Now we have 4 jobs left unassigned. So let's look at the 2nd machine. For that machine, any of the 4 remaining jobs may be assigned to it, leaving 3 unassigned jobs. We can continue in that fashion, assigning at random one the of 3 remaining jobs to the 3rd machine, one of the 2 remaining jobs to the 4th machine, and finally, the only unassigned job to the 5th machine. So there's 5 * 4 * 3 * 2 * 1 = 5! = 120 different ways to assign those 5 jobs to all 5 machines.