## Ask Professor Puzzler

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Mathi, from Vellore, wants to know how to figure out the following game show probability:

"On a game show, a contestant is given three keys, each of which opens exactly one of three identical boxes. The first box contains $1, the second $100, and the third $1,000. The boxes are randomly lined up and the contestant gets to assign each key to one of the boxes. The contestant wins the amount of money contained in each box that is opened by the key he assigns to it. What is the probability that a contestant will win more than $1,000?"

I'm not going to answer that exact question, because I think we can make it more challenging and interesting by changing the numbers a bit. Let's do this one instead:

"On a game show, a contestant is given four keys, each of which opens exactly one of four identical boxes. The first box contains $250, the second $500, the third $750, and the fourth $1,000. The boxes are randomly lined up and the contestant gets to assign each key to one of the boxes. The contestant wins the amount of money contained in each box that is opened by the key he assigns to it. What is the probability that a contestant will win more than $1,000?"

First, we need to figure out how many ways the keys can be arranged. The first key can be assigned in 4 ways, the second one in 3 ways (since one key has already been placed), the third key in 2 ways, and then there's only one way to place the last key. That gives us a total of 4 x 3 x 2 x 1 = 24 ways. So if we can figure out how many possibilities are wins, all we need to do is divide that by 24 to get the answer.

It's a win if you place all the keys in the right position. There is **one **way to do that.

It's a win if you place $1000 and any *one* of the others correctly. (Note that you can't place $1000 and *two *others correctly, because if you place three of them correctly, the fourth one must be correct as well, and we've already counted that possibility!). So there are **three **ways to do that.

It's a win if you place the $500 and $750 correctly (but not the other two, since we've already counted that possibility!). You can do that in **one **way.

And there are no other combinations that work. Thus, we have a probability of (1 + 3 + 1)/24 = 5/24.

Now that we've gone through that, you'll be surprised at how easy the other problem is - there are far fewer combinations to consider!

Good luck!

Professor Puzzler