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Plus Advent Calendar Door #6: Six out of forty-nine

Now here's an important mathematical question: how likely are you to win the lottery?

1 in 14 million.

In the UK lottery you have to choose 6 numbers out of 49, and for a chance at the jackpot you need all of your 6 numbers to come up in the main draw. So the question is really how many possible combinations of 6 numbers can be drawn out of 49? There are 49 possibilities for the first number, 48 for the second, and so on, to 44 possibilities for the sixth number, so there are 49x48x47x46x45x44=10,068,347,520 ways of choosing those six numbers... in that order.

But we don't care which order our numbers are picked, and the number of different ways of ordering 6 numbers are 6x5x4x3x2x1=6! =720: there are six possibilities for the number that comes first in the list, 5 for the number that comes second, and so on. Therefore our six numbers are one of 49x48x47x46x45x44/6! = 13,983,816 so we have about a one in 14 million chance of hitting the jackpot. Hmmm...

But on a brighter note, we have just discovered a very useful mathematical fact: the number of combinations of size $k$ (sets of objects in which order doesn’t matter) from a larger set of size $n$ is:

  \[ \frac{n!}{(n-k)!k!}, \]    

where

  \[ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1. \]    

This sort of argument lies at the heart of combinatorics, the mathematics of counting. It might not help you win the lottery, but it might keep you healthy. It is used to understand how viruses such as influenza reproduce and mutate, by assessing the chances of creating viable viruses from random recombination of genetic segments.

You can read more on the uses of combinatorics, including in money matters (lottery), love (well, kissing frogs) and fun (juggling and Rubik's cubes):

Combinatorics on Plus

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