P&C

Gordon buys 5 dolls for his 5 nieces. The gifts include two identical Sun-and-Fun
beach dolls, one Elegant Eddie dress-up doll, one G.!. Josie army doll, and one Tulip
Troll doll. If the youngest niece does not want the G.!. Josie doll, in how many different
ways can he give the gifts?

The solution below assumes that every niece is to receive exactly 1 gift.
Any ARRANGEMENT of the 5 gifts represents ONE WAY to distribute the gifts.

Good = Total - Bad.

Total possible arrangements:
Number of ways to arrange the 5 gifts = 5! = 120.
But 2 of the dolls are IDENTICAL.
When the identical dolls swap positions, the arrangement doesn't change, reducing the number of unique arrangements.
To account for the smaller number of unique arrangements, we must divide by the number of ways to ARRANGE the 2 identical dolls:
5!/2! = 60.

Bad arrangements:
In a bad arrangement, the youngest niece receives the GI Josie doll.
Number of options for the youngest niece = 1. (The GI Josie doll.)
Number of ways to arrange the 4 remaining gifts = 4!/2!. (We divide by 2! to account for the 2 identical dolls.)
To combine these options, we multiply:
1 * 4!/2! = 12.

Thus:
Good arrangements = 60-12 = 48.