Lottery Combinations Homework: Solving for Guaranteed Matches

[jumbogala]

Homework Statement

Students are playing a lottery game. In this game, three numbers are drawn from a set of six.

If the three numbers on a student's ticket match those drawn, the student wins the full prize. If just one or two numbers match, the student wins a consolation prize.

The order of the numbers doesn't matter. Also, once a number is picked it cannot be picked again.

a) How many tickets would a student have to buy to guarantee that at least one number out of 3 matches the winning combination?

b) What if at least 2 numbers need to match?

c) What if all 3 numbers need to match?

Homework Equations

The Attempt at a Solution

a) I know that the answer to is 2 tickets. This is because you could buy the following:

1,2,3 on one ticket. 4,5,6 on the other. This way, at least one of the numbers will match the winning ticket.

But I have to represent this mathematically. Would it be (6 choose 1) / 3?

b) If my formula is correct, then it would be (6 choose 2) / 3

c) (6 choose 3) / 3

But I am not sure if my formula is right. I'm having trouble reasoning it out in my head.

 

[boaz]

It seems fine to me, except the last one. In general, you get the total number of combinations and divide it by the number of winning combinations in each ticket, as a result you get the number of tickets.

For instance, in the 2nd question, we have C(6,2) combinations. However, each ticket has 3 combination (because if we have x,y,z then (x,y), (y,z) and (x,z) are winning combinations). However, in the 3rd question we have C(6,3) but each ticket covers only one winning combination.