Problem 233

Solution 1:

We begin by naming the $2$
quarters Q1 and Q2, the $2$ dimes D1
and D2, and the $2$ nickels N1 and
N2.

There are $6$ possible choices for
the first coin, followed by $5$
choices for the second coin. However, this double counts the number of
possibilities since choosing for example Q1 first and D2 second is the
same as choosing D2 first and Q1 second, and so there are a total of
$\dfrac{6\times5}{2}=15$ ways to
choose the $2$ coins.

Next, we count the number of ways to obtain a combined value of at least
$\$0.30.$

These are: Q1 and Q2 $(\$0.50)$, Q1
and D1 $(\$0.35)$, Q1 and D2 $(\$0.35)$, Q1 and N1 $(\$0.30)$, Q1 and N2 $(\$0.30)$, Q2 and D1 $(\$0.35)$, Q2 and D2 $(\$0.35)$, Q2 and N1 $(\$0.30)$, and finally Q2 and N2 $(\$0.30)$. This gives a total of $9$ different ways.

The probability that the combined value of the two coins is $\$0.30$ or more is $\dfrac{9}{15}=\dfrac{3}{5}$.

Solution 2:

The probability that the combined value of the two coins is at least
$\$0.30$ is equal to the sum of the
probabilities of the combined values being equal to $\$0.30$, $\$0.35$, and $\$0.50$.

The probability that the two coins have a combined value of $\$0.30$ is equal to the probability of
choosing a quarter followed by a nickel, or choosing a nickel followed
by a quarter. The probability that the first coin chosen is a quarter is
$\dfrac26$, and with $5$ coins remaining, the probability that
the second coin chosen is a nickel is $\dfrac25$. Thus, the probability of
choosing a quarter followed by a nickel is $\dfrac26\times\dfrac25=\dfrac{4}{30}$. The
probability of choosing a nickel first followed by a quarter second is
also equal to $\dfrac{4}{30}$, and so
the probability that the combined value of the two coins is $\$0.30$ is $\dfrac{4}{30}+\dfrac{4}{30}=\dfrac{8}{30}$.

The probability that the two coins have a combined value of $\$0.35$ is equal to the probability of
choosing a quarter followed by a dime, or choosing a dime followed by a
quarter. As in the previous case, the probability that the combined
value of the two coins is $\$0.35$ is
also $\dfrac{4}{30}+\dfrac{4}{30}=\dfrac{8}{30}$.

The probability that the two coins have a combined value of $\$0.50$ is equal to the probability of
choosing a quarter followed by a quarter. The probability of choosing a
quarter followed by a quarter is $\dfrac26\times\dfrac15=\dfrac{2}{30}$.

Thus, the probability that the combined value of the two coins is at
least $\$0.30$ is equal to $\dfrac{8}{30}+\dfrac{8}{30}+\dfrac{2}{30}=\dfrac{18}{30}=\dfrac35$.