Multiplicative counting - the different types

In preparing for teaching the new GCSE, one of the new topics is the explicit use of multiplicative counting. While preparing problems I came to a realisation that there are 3 different types of counting here (at least before we get into the whole permutations and combinations area), and I thought I would take this opportunity to share my insight.

1) Powers for counting

Students and teachers of A-Level stats modules will likely be intimately familiar with this idea through the binomial theorem. This occurs when the same choice or outcome is independently repeated. The typical example might be flipping coins; flipping 1 coin has 2 outcomes, flipping 2 coins has 4 outcomes, flipping 3 coins has 8 outcomes etc, leading to 2ⁿ for the possible different outcomes of n coins. These are probably the easiest type to make up as there just needs to be a repeat that isn't affected by the outcomes before. Tweaking independent event probability questions to focus on the outcomes can be a rich source of ideas here.

2) Multiplying for counting

This one is probably the most familiar to GCSE teachers, as it is a reasonably well used stretch for pupils on systematic listing. Similar to above, this occurs when choices are independent, but when there are different number of choices for each outcome. Probably the most typical here is the menu problem: choose a three course meal from 3 starters, 4 main courses and 3 desserts and there are
3 x 4 x 3 = 36 different meals you can choose from. Take any systematic listing problem from the current GCSE and it can be tweaked to encourage multiplicative counting; just change the question from "list all the possible outcomes" to "work out how many possible outcomes there are" and maybe add a few more options to make the multiplicative reasoning justifiable over just creating a list in the first place.

3) Factorial counting.

This is perhaps the one that will be least familiar, although still familiar to the A-Level aficionados amongst us; this occurs when an outcome is removed once used and the process repeated until no outcomes are left. A nice example of this is given here:

Anne, Barry, Colin and Damien book 4 seats next to each other in the cinema as shown.

Seat 1

Seat 2

Seat 3

Seat 4

One way that they could sit with each other is like this:

Seat 1	Seat 2	Seat 3	Seat 4
Anne	Barry	Colin	Damien

(a) How many different ways can they sit?

The result here is 4 x 3 x 2 x 1 = 4! = 24, because there are 4 places the first person could sit, then once settled there are 3 places for the second person, then 2 for the third, then only 1 for the fourth (or alternatively, 4 options for the first seat, then 3 options for the second seat etc...). A nice place to go from here is to ask 

(b) What is the probability that Anne and Barry sit next to each other?.

This is a fairly simple adaptation to the problem, which could be asked directly instead of breaking down the problem into part (a) and part (b), but importantly it leads to a further question:

(c) How is your answer to part (b) affected if the four seats are not next to each other in a cinema, but around a table on a train as shown?

Seat 1

Seat 2

Seat 4

Seat 3

This sort of variation isn't really available to the first two problems, but is a lovely adjustment in these sort of factorial counting problems. Although they are harder to come up with, modifications of problems with conditional probability can be fruitful here, often with the reduction of options (i.e. going from choosing 2 or 3 from a bag of 20 sweets to eating 4 or 5 sweets in a certain order).

I guess I have two points from this blog. The first would be to make sure that pupils are aware of these different approaches to counting, and the second is to suggest that you don't need to come up with lots of new contexts for this new topic, just tweak things that are already there.