Lottery probability question... A person participates in a state lottery by selecting six numbers from 1 through 59. If the six numbers match the six drawn by the lottery, regardless of order, then the participant wins the first prize of millions of dollars. If a participant's numbers match five of the six drawn, the participant wins second prize, which is not millions of dollars.
I know that the probability of winning the first prize is .0000000222 because of 1 / (59 C 6).
How can I find the second prize winner?

The Corporation must pick exactly $5$ of our numbers. Which $5$? These can be chosen in $\binom{6}{5}$ ways. 
For each of these ways, there are $\binom{53}{1}$ ways for the Corporation to choose a sixth number that doesn't match  ours. 
It follows that there are $\binom{6}{5}\binom{53}{1}$ different Corporation choices that give us a  second prize.
For the probability, divide by $\binom{59}{6}$. 
