How do I do this problem?

Question

In one lottery, a player wins the jackpot by matching all 5 distinct numbers drawn in any order from the white balls (1-42) and matching the number on the gold ball (1-33). If 1 ticket is purchased, what is the probability of winning the jackpot? Type an integer or a simplified fraction.

Answer 1

First ask yourself, how many possible tickets are there? We are selecting 5 distinct numbers from a set of 42, and we are selecting 1 number from a set of 33. How many ways are there to do the first selection, where we draw 5 numbers from a set of 42? That would be 42 choose 5, or 42!/(5!37!), which works out to 850,668. Next, how many ways are there to do the second selection, where we draw 1 number from a set of 33? This one is a little easier: 33 choose 1, or 33!/(1!32!), which is simply 33.

So, what is the total number of ways to select these numbers? Since there are 850,668 ways to do the first selection, and 33 ways to do the second selection, we have 850,668*33 = 28,072,044 total ways to draw five numbered white balls plus one numbered gold ball. Since the ticket we purchase is equally likely to have any one of those 28,072,044 options, and only one of them actually wins, the probability of winning is 1/28,072,044.