Question

Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical Sun-and-Fun beach dolls, 1 Elegant Eddie dress-up doll, 1 G.I. Josie army doll, and 1 Tulip Troll doll. If the youngest niece does not want the G.l. Josie doll, in how many different ways can he give the gifts?

• A. $50$
• B. $48$
• C. $54$
• D. $60$

Answer 1

Answer: (B)

$48$

First, solve the problem without considering the fact that the youngest girl does not want the G.!. Josie doll.

Gordon's nieces could get either one of the Sun-and-Fun dolls, which we'll call S, or they could get the Elegant Eddie doll (E), the Tulip Troll doll (T), or the G.I. Josie doll (G). This problem can be modeled with anagrams for the "word" SSETG:

$\frac{5!}{2!}=5\times 4\times 3=60$

Note that you should divide by 2! because of the two identical Sun-and-Fun dolls.

Thus, there are 60 ways in which Gordon can give the gifts to his nieces.

However, you know that the youngest girl (niece E) does not want the G.I. Josie doll. So, calculate the number of arrangements in which the youngest girl does get the G.I. Josie doll. If niece E gets doll G, then you still have 2 S dolls, 1 E doll, and 1 T doll to give out to nieces A, B, C, and D. Model this situation with the anagrams of the "word" SSET:

$\frac{4!}{2!}=12$

There are 12 ways in which the youngest niece will get the G.I. Josie doll.

Therefore, there are 60 - 12, or 48, ways in which Gordon can give the dolls to his nieces.