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 nn people in a straight line, we can put nn people in the first position, n1n-1 people in the second position, n2n-2 in the third, and so on. Overall, then the number of ways of arranging distinct people in a line is equal to

(n)(n1)(n2)1=n!(n)(n-1)(n-2)...1=n! (18.1)


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 AA or in way BB

number of ways for AA + number of ways for BB

To have "whatever" I need both AA and then BB

number of ways of AA ×\times number of ways BB

For every item in this set of nn objects there are kk ways of obtaining it.


I have a group of nn things, and I want to pick kk of them, but I don’t care about the order in which I get them.


I have a group of nn different things, and I want to pick kk of them, and I do care about the order in which I get them


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.). 11 1 This strategy works really well throughout mathematics, but it’s especially helpful here.