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You know those lottery games where you pick five or six numbers from 1 to 42, or 1 to 55, and if you pick the same numbers that the lottery does, you win some obscene amount of money? A couple of years ago, my wonderful gamblin' grandma bought me a season ticket to a game here in Massachusetts called Megabuck$ (the dollar sign emphasizes that there is big money to be won here, folks). She was a steady long-time lotto player and won some big prizes over the years (though she never won the jackpot). After she got me the ticket, naturally I wanted to know what my odds of winning were.

Like most people, at first I thought this was pretty easy to figure out. You pick six numbers from 1 to 42. So it's just 42 * 41 * 40 * 39 * 38 * 37. Simple. And that puts my odds of winning at ... whoa! 1 in 3,776,965,920! Damn, Admiral Megabuck$. Those are some harsh odds. I logged on to the Mass Lottery site to confirm what I had calculated -- and was confused to see that my odds of winning were actually quite a lot better: 1 in 5,245,786. What gives? Was the Mass State Lottery lying to me?

I continued to live my life in utter confusion until last year, when I took a class in discrete math, and learned that whether a set of numbers is ordered makes a big difference in determining all the possible combinations. If the numbers are ordered, then you use the formula from above. If they're not, you use a different formula, which is the one that the Lottery used. But I still couldn't figure it out. The Lottery numbers are ordered, aren't they? I mean, it's a sequence of numbers, for God's sake! How could they not be ordered?

A few nights ago I started thinking about the problem again, and it finally dawned on me: the numbers are not ordered -- they only appear to be. The whole setup of picking the numbers -- even having the numbers printed on the ping pong balls -- distract you from the fact that they're not ordered. Oh yes, Admiral Megabuck$ is a sly fellow. But he is also fair.

A Simpler Example for Simple People (like me)

To illustrate, let's forget about the balls and look at a smaller and more personal example. Suppose we have a group of five people:

  • Karen
  • Rudolph
  • Alf
  • Francis
  • Ruttiger

What are the odds that if you pick three people, and I pick three people, that we'll both pick the same three people? Well, if I pick Karen, Rudolph and Ruttiger, in that order, and you pick Rudolph, Karen and Ruttiger, in that order, do we have a match? Yes. Although we picked them in a different order, we still picked the same three people: clearly, a group made up of three distinct people is the same group of three distinct people, no matter how you order them.

The same thing is going on with Megabuck$. It doesn't matter which numbers get picked first, or which get picked last. Once six numbers have been chosen, they're put into ascending order, and that is your winning number. So if the lovely and talented Dawn Hayes calls out 1 / 37 / 19 / 22 / 41 / 9, it's the same as if she called out 37 / 41 / 22 / 9 / 1 / 19 / 41, or 9/ 1 / 19 / 22 / 37 / 41, or any other ordering of those six numbers. This is why the lotto numbers are unordered.

As a counter example, consider license plate numbers, which are ordered. AZB294 is not the same as 294BAZ.

Doing the Maths

So now that we know this, how do you figure out the odds? Here's the formula:

Formula for combinations

Where n is the number of things you're choosing from and r is the number of things you're choosing. So we just plug in our numbers for Megabuck$:

Formula for Megabuck$

Which gives us 5,245,786. And there you have it.