Myriora Maths › 11+ maths › Chapter 3: Counting and logic

Listing systematically

11+Chapter 3 · Lesson 3

In this lesson

List in order, starting with the smallest, so you don't miss any or repeat any.
If every first choice has the same number of next choices, multiply.
And if the order doesn't matter, don't count the same pair twice.

Your turn

Using the digits one, two, three and four, how many two-digit numbers can you make, with two different digits?

Show the answer

There are four choices for the tens digit. Then three choices are left for the ones digit. Four times three is twelve. Here they all are, listed in order!

The common mistake

A common mistake is counting pairs twice. Four flavours, times three flavours left, is twelve. But that counts A B and B A as different, when they're the same ice cream! Twelve divided by two is six.

Try it at home

Count outfits from real tops and trousers, or make every number from three number cards.

Read the whole lesson

Hello, and welcome to Myriora Maths! How many different numbers can you make with the digits one, two and three, using each digit once? If you just guess, you'll probably miss some. Let's be systematic!

Start with the smallest first digit: one. The other two digits can go either way: one two three, and one three two. Next, start with two: two one three, and two three one. Then start with three: three one two, and three two one. That's six numbers, in order from smallest to biggest.

A tree diagram shows the same thing. There are three choices for the first digit. Then two choices for the second digit. Then just one digit is left for the last place. Three times two times one is six.

Listing works for all sorts of choices. You have three tops and two pairs of trousers. Each top goes with each pair of trousers. So there are three times two: six outfits.

But be careful when the order doesn't matter. Choose two ice cream flavours out of four: A, B, C and D. List them in order: A B, A C, A D, then B C, B D, then C D. That's six pairs. B A is the same as A B, so we don't count it again!

A common mistake is counting pairs twice. Four flavours, times three flavours left, is twelve. But that counts A B and B A as different, when they're the same ice cream! Twelve divided by two is six.

Your turn! Using the digits one, two, three and four, how many two-digit numbers can you make, with two different digits? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! There are four choices for the tens digit. Then three choices are left for the ones digit. Four times three is twelve. Here they all are, listed in order!

Quick review. List in order, starting with the smallest, so you don't miss any or repeat any. If every first choice has the same number of next choices, multiply. And if the order doesn't matter, don't count the same pair twice. See you next time!