Question

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?

Answer 1

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.

Answer 2

The fundamental principle of counting is important to solve this particular problem. For the first person, we are able to choose from 4 men, for the second in line, we can choose from 4 women. When alternate, the counting becomes,

4 x 4 x 3 x 3 x 2 x 2 x 1 x 1

or it can be simplified to

4! x 4!

Therefore, the answer is 576.