Imagine arranging 6 people to sit on a bench and 3 of them have to sit side by side. How many relevant arrangements will be made?

Consider the relative positions of the 3 people. Let's say the 3 people who have to sit together are A, B, and C. We can think of the different arrangements as follows:
A B C (all 3 are together)
A C B
B A C
B C A
C A B
C B A
There are 6 ways to arrange the 3 people within their unit. But for each of these arrangements, the 3 people can also be switched around within their unit, so the total number of ways is 6 × 3! = 144.