# 18.1 Basic counting principles

## 18.1.1 Finding the number of permutations

How many ways can we arrange distinct people in a straight line? For $n$ people in a straight line, we can put $n$ people in the first position, $n-1$ people in the second position, $n-2$ in the third, and so on. Overall, then the number of ways of arranging distinct people in a line is equal to

TODO

## 18.1.2 From English to maths

One thing that can be tricky in combinatorics is working out what words in English mean in terms of combinatorial operations. Here’s a handy dictionary.

English | Combinatorics |

This can happen in way $A$ or in way $B$ |
number of ways for $A$ + number of ways for $B$ |

To have "whatever" I need both $A$ and then $B$ |
number of ways of $A$ $\times$ number of ways $B$ |

For every item in this set of $n$ objects there are $k$ ways of obtaining it. |
$k^{n}$ |

I have a group of $n$ things, and I want to pick $k$ of them, but I don’t care about the order in which I get them. |
$\binom{n}{k}$ |

I have a group of $n$ different things, and I want to pick $k$ of them, and I do care about the order in which I get them |
$\frac{n!}{(n-k)!}$ |

One strategy I find very useful in solving combinatorics problems is to write
out a description of what I’m after in English, and then translate this into
combinatorial operations (e.g. permutations, combinations, etc.).
^{1}^{1}
1
This strategy works really well throughout mathematics, but it’s
especially helpful here.