Suppose you have $5$ boxes, each of which contains a red ball, a blue ball, a green ball, a yellow ball and a white ball.
If you draw a ball at random from each box, what is the probability that there are exactly two balls of the same color?
Stuck a small amount with this question:
The probability of getting two balls of the same color is
$$P(\text{$2$ Same Color}) = 1 \cdot \frac{1}{5} \cdot \frac{4}{5} \cdot \frac{3}{5} \cdot \frac{2}{5} = \frac{24}{625}$$
Then just multiplying $P(\text{$2$ Same Color})$ by $5$ as there are $5$ different colors of balls so you could get $2$ whites or $2$ reds etc,
That gives you
$$\frac{24}{625} \cdot 5 = \frac{24}{125}$$
Not sure what the answer to this is just looking for someone to tell me if I'm on the right track or completely wrong.
Best Answer
Pick the color of the pair ($5$) then choose the two boxes that you draw that color from ($_5C_2 = 10$). Then, pick three colors other than the pair's color $(_4C_3 = 4)$ and choose the order that those colors are picked out of the other three boxes $(3! = 6)$.
That gives $1200$ ways to pick exactly one pair. Divide by $5^5 = 3125$ ways to pick any five balls, and you have your answer.