Problem Solving

Any 5-letter word we make using the letters S, S, E, G, T corresponds to one way to distribute the dolls. So the word SEGTS for example corresponds to giving the S doll to the oldest child, the E to the next oldest, and so on.

If the letters were different, there would be 5! words we could make using 5 letters. Because the order of the two S's doesn't matter, we need to divide by 2!, so there are 5!/2! = 60 words we can make. But in exactly 1/5 of those words, the letter G is the last letter, so the youngest child is getting the G doll in 12 of those words. Assuming that's not allowed (something the question should actually say), there are 60-12 = 48 ways to distribute the dolls in total.

AAPL wrote:Manhattan Prep

Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn't want the G doll, in how many different ways can he give the gifts?

A. 12
B. 24
C. 36
D. 48
E. 60

OA D

Since the youngest niece doesn't want the G doll, she has 4 choices of the dolls. The other 4 nieces will then have 4, 3, 2 and 1 choices for the remaining dolls. Therefore, the "number" of ways to distribute the dolls is:

4 x 4 x 3 x 2 x 1 = 16 x 6 = 96

However, since 2 identical S beach dolls are indistinguishable, like permutation of indistinguishable objects, we have to divide by that number factorial. Therefore, the actual number of ways to distribute the dolls is:

96/2! = 96/2 = 48

Alternate Solution:

Let's first calculate the number of ways to distribute the dolls without the youngest niece constraint. We have 5 dolls, 2 of which are identical; therefore, by the permutation of indistinguishable objects formula, there are

5!/2! = (5 x 4 x 3 x 2)/2 = 60

ways to distribute, if we ignore the youngest niece getting the G doll.

Let's now determine how many of these 60 ways assign the G doll to the youngest niece. If we assume the youngest niece did get the G doll, we now have 4 dolls, 2 of which are identical. Therefore, there are

4!/2! = (4 x 3 x 2)/2 = 12

ways where the youngest niece gets the G doll.

Thus, the dolls can be distributed in 60 - 12 = 48 ways where the youngest niece does not get the G doll.

Answer: D