Maths in a minute: Combinatorics

How likely are you to win the lottery?

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 $49 \times 48 \times 47 \times 46 \times 45 \times 44 = 10068347520$ 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 picking 6 numbers are $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 6! = 720$. Therefore our six numbers are one of $49 \times 48 \times 47 \times 46 \times 45 \times 44 / 6! = 13983816$ 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 $n!(n−k)!k!.$

This sort of argument lies at the heart of combinatorics, the mathematics of counting. It might not help you win lotto, 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.

Πηγή: plus.maths