19. A random number generator gives outputs of $1,2,3,4$ and 5 with equal probability. The values of $a, b$ and $c$ are each chosen by running the generator once.
The probability that $a \times b+c$ is even can be written as a fraction in its lowest terms as $\frac{N}{D}$.
What is the value of $10 N+D$ ?
Solution
715

The expression $a \times b+c$ is even if $a b$ and $c$ are either both odd, or both even.
The product $a b$ is odd only if $a$ and $b$ are both odd. This occurs with probability $\frac{3}{5} \times \frac{3}{5}=\frac{9}{25}$. Therefore the probability that both $a b$ and $c$ are odd is $\frac{9}{25} \times \frac{3}{5}=\frac{27}{125}$.
The product $a b$ is even with probability $1-\frac{9}{25}=\frac{16}{25}$.
Therefore the probability that both $a b$ and $c$ are even is $\frac{16}{25} \times \frac{2}{5}=\frac{32}{125}$.
The total probability that $a b+c$ is even is $\frac{27}{125}+\frac{32}{125}=\frac{59}{125}$.
Therefore $N=59, D=125$ and $10 N+D=10 \times 59+125=715$.