The product rule for counting (Higher)

You are on a website building your own trainers. A base colour. A stripe colour. Laces. In the corner, a counter says: two hundred and forty designs available. Nobody typed out two hundred and forty designs. The site got that number in one line of arithmetic. By the end of this video, so will you.

One thing before we start. This is a Higher tier topic. If you are sitting Foundation, you have not missed anything here. Your paper stops at listing.

This is the final video in our chapter on factors, multiples and counting.

The video on systematic listing shows the method. Fix one thing, cycle the other. Keep that method. This is where it runs out of road. A phone case shop. Six colours. Four patterns. One colour and one pattern make a design. How many different designs are possible? List it exactly as before. Fix red. Red stripes. Red spots. Red plain. Red marble. Four designs, and red is finished. Fix black. Stripes, spots, plain, marble. Four again. Not roughly four. Exactly four, every single time, because the patterns do not care which colour they sit on.

So you can finish this without writing another line. Six colours, four patterns each. Is the total ten, twenty four, or thirty six?

Pick one. I'll give you a few seconds.

The answer is twenty four. Now look at why. The finished list is six blocks, one block per colour, and every block holds exactly four.

You are not multiplying because a rule tells you to. You are multiplying because the list is identical blocks stacked up, and counting identical blocks is what multiplying is. It has a name. The product rule for counting.

Back to the trainers. Ten base colours, six stripes, four laces. Take the first two. Ten blocks of six is sixty colour and stripe pairs. Now add the laces. Every one of those sixty pairs can take any of the four. So the list of sixty becomes four copies of itself. Sixty times four. Two hundred and forty.

Keep that sentence. A new stage does not add to your list. It copies it. That is why these totals climb so fast, and why listing gives up so early.

Now some exam craft. Here is what an examiner wrote about a counting question on a real paper. Part a was not well answered. The product rule for counting was often not applied at all with many students attempting to list all the possible codes, with little success. Not a wrong method. No method. The listing instinct, on a question far too big to list.

So, that question rebuilt. A padlock code has three digits, each one from zero to nine, but the first digit cannot be zero. How many codes are possible? Count the digits properly first. Zero, one, two, three, four, five, six, seven, eight, nine. That is ten digits, not nine. Zero counts.

Three stages, ten digits each. But the first digit cannot be zero. How many choices does the first digit have. Nine, or ten?

Your call. I'll give you a moment.

The answer is nine. Zero is barred, so that stage loses one option and keeps the other nine.

Here is the careful part. The restriction named the first digit only. The second and third digits were never mentioned, so they still have all ten. Nine, ten, ten. Nine times ten is ninety. Ninety times ten is nine hundred. Nine hundred possible codes. The same report notes that some students who did use the rule still lost part of the restriction. Ignore it and you get a thousand. Spread it over all three stages and you get seven hundred and twenty nine. Both wrong, from one careless read. So before you multiply, read the question again, hunting only for the words that cut choices down. Cannot be. Must be. Only. Change the count at the stage those words name, and leave the others alone.

Your turn. A t-shirt comes in five colours and three sizes. How many different options are there? Do it without writing out a single combination.

I'll wait, then work it through with you.

Two stages. Five times three. Fifteen options. Five blocks of three, or three blocks of five. Either picture gives you the same fifteen.

So, let's pull it together. Count the stages. Count the choices at each stage. Multiply straight across. A new stage copies your list, it does not lengthen it. And a restriction changes the count at the one stage it names.

And that completes our chapter on factors, multiples and counting. If you are following the course, the next chapter is powers, roots and standard form.

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