I'm trying to solve the following problem:
In a fraternity house, three boys share a room with a single closet. Each boy can wear each of the other boys items of clothing, and they share freely. The closet contains 3 pairs of shoes, 7shirts, 5 pairs of pants, 8 pairs of socks and 4 coats. If each boy dresses in shoes, shirt, pants, socks, and coat, in how many combinations of clothing may the boys appear together?
This is from Hays, 1973.
In the end it gives the solution: 3*7!8! or (7!)(8!)/ (3!)ˆ2*2!
I've spent the last 15 minutes trying to figure out where all the other items disappeared. I thought the answer would be 3!*7!*5!*8!.
Where am I going wrong?