Problem 432

The probability that all $3$
socks are not the same colour is equal to $1$ minus the probability that all $3$ socks are the same colour.

If all $3$ socks are the same colour,
then they must all be black or they must all be gold (they cannot all be
white since there are only $2$ white
socks).

We begin by determining the probability that all $3$ socks are black.

There are $5+3+2=10$ socks in total,
and $5$ of these are black.

Therefore, the probability that the first sock that Jack removes is
black is $\dfrac{5}{10}$.

There are now $9$ socks in the drawer
and $4$ of them are black, and so the
probability that the second sock removed is black is $\dfrac{4}{9}$.

The probability that the third sock removed is black is $\dfrac{3}{8}$, and so the probability that
all $3$ socks are black is $\$\dfrac{5}{10}\times \dfrac{4}{9}\times \dfrac{3}{8}=\dfrac{60}{720}\$.$

Similarly, the probability that all $3$ socks removed are gold is $\$\dfrac{3}{10}\times \dfrac{2}{9}\times \dfrac{1}{8}=\dfrac{6}{720}\$.$

Thus the probability that all $3$
socks removed are the same colour is $\dfrac{60}{720}+\dfrac{6}{720}=\dfrac{66}{720}=\dfrac{11}{120}$,
and so the probability that all 3 socks removed are not the same colour
is $1-\dfrac{11}{120}=\dfrac{109}{120}$.