A driver approaching a toll booth has exactly two quarters, two dimes and two nickels in his pocket. He reaches into his pocket and randomly selects two of these coins. What is the probability that the coins that he selects will be at least enough to pay the 30 -cent toll?
(A) $\frac{3}{5}$
(B) $\frac{2}{5}$
(C) $\frac{1}{3}$
(D) $\frac{3}{10}$
(E) $\frac{2}{3}$
We start by labelling the two quarters $Q_{1}$ and $Q_{2}$, the two dimes $D_{1}$ and $D_{2}$, and the two nickels $N_{1}$ and $N_{2}$. We then make a chart where the labels on the left tell us which coin is chosen first and the labels on the top tell us which coin is chosen second. Inside the chart, we put a Y if the combination will pay the toll and an $\mathrm{N}$ if the combination will not pay the toll. (We put X's on the diagonal, since we cannot choose a coin first and then the same coin the second time.)

|  | $Q_{1}$ | $Q_{2}$ | $D_{1}$ | $D_{2}$ | $N_{1}$ | $N_{2}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $Q_{1}$ | $\mathrm{X}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{Y}$ |
| $Q_{2}$ | $\mathrm{Y}$ | $\mathrm{X}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{Y}$ |
| $D_{1}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{X}$ | $\mathrm{N}$ | $\mathrm{N}$ | $\mathrm{N}$ |
| $D_{2}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{N}$ | $\mathrm{X}$ | $\mathrm{N}$ | $\mathrm{N}$ |
| $N_{1}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{N}$ | $\mathrm{N}$ | $\mathrm{X}$ | $\mathrm{N}$ |
| $N_{2}$ | $\mathrm{Y}$ | $\mathrm{Y}$ | $\mathrm{N}$ | $\mathrm{N}$ | $\mathrm{N}$ | $\mathrm{X}$ |

For example, if $Q_{1}$ is chosen first and then $N_{1}$, the driver has chosen 30 cents, so can pay the toll. If $D_{2}$ is chosen first and then $D_{1}$, then 20 cents has been chosen, and he cannot pay the toll.

From the chart, there are 30 possible combinations that can be chosen (since the X's on the main diagonal indicate that these combinations are not possible), with 18 of them enough to pay the toll, and 12 not enough to pay the toll.

Therefore, the probability is $\frac{18}{30}=\frac{3}{5}$.