IB Maths AA 1.8 Notes

Counting principles

This section extends earlier work on combinations and binomial expansion. It introduces counting principles, permutations and combinations, then extends the binomial theorem to cases where the index is negative or fractional.

Counting principles help us work out how many possible outcomes there are in a situation. The most important rule is the fundamental counting principle.

If one task can be done in $m$ ways and a second task can be done in $n$ ways, then the two tasks together can be done in $m\times n$ ways.

This is the AND rule, so separate stages are multiplied. If there are different choices that cannot happen together, then the total number of possibilities is found by adding. This is the OR rule.

Determine how many outfits can be made from $4$ shirts AND $3$ pairs of trousers, and $4$ shirts OR $3$ pairs of trousers

If there are $4$ shirts and $3$ pairs of trousers, then the number of outfits with one shirt and one pair of trousers is $4\times3=12$.

If there are $4$ shirts or $3$ pairs of trousers to choose from, then the number of choices is $4+3=7$.

Let's put this to the test:

A shop sells $4$ shirts, $8$ pairs of shoes and $2$ pairs of glasses. How many ways can one choose a shirt and either shoes or glasses?

First count shirt and shoes: $4\times8=32$.

Then count shirt and glasses: $4\times2=8$.

Now add because it is 'either shoes or glasses': $32+8=40$.

So there are $40$ possible choices.