The letters G, A, U, S, and S are written on five tiles, one letter per tile. If Amy selects two tiles at random, what is the probability she gets two $S$ 's?
(A) $\frac{3}{5}$
(B) $\frac{2}{5}$
(C) $\frac{1}{8}$
(D) $\frac{1}{10}$
(E) $\frac{1}{20}$

## Part C: Each correct answer is worth 8.
This question is made more difficult by the fact that there are 2 letters that are the same. To overcome this problem, let us change the second $\mathrm{S}$ to a $\mathrm{T}$, so that the letters on the tiles are now $\mathrm{G}, \mathrm{A}, \mathrm{U}, \mathrm{S}$, and $\mathrm{T}$, and we want to calculate the probability that Amy chooses the $\mathrm{S}$ and the $\mathrm{T}$ when she chooses two tiles.

Let us say that Amy chooses the tiles one at a time. How many choices does she have for the first tile she chooses? Since there are 5 tiles, she has 5 choices. How many choices does Amy have for the second tile? There are 4 tiles left, so she has 4 choices. Now for each of the 5 ways of choosing the first tile, she has 4 ways of choosing the second tile, for a total of 20 possibilities. (Try writing out the 20 possibilities and convince yourself that it is $5 \times 4$ and not $5+4$.)

In these 20 pairs, there are two pairs that contain an $S$ and $T$. If the $T$ was changed back to an $S$, there would be two pairs out of 20 containing $2 \mathrm{~S}$ 's so the probability would be $\frac{2}{20}=\frac{1}{10}$ of selecting the 2 S's.

ANSWER: (D)

## Part C