Four men and four women line up at a checkout counter in a store. in how many different ways can they line up if the first person in line is a man, and the people in line alternate man, woman, man, woman, and so on?

To answer the question, make use of the Fundamental Principles of Counting. For the first person in the line, we can choose from 4 men, for the second person from 4 women. Alternately, the counting becomes,
                     4 x 4 x 3 x 3 x 2 x 2 x 1 x 1      or     4! x 4!
The answer to problem is 576.