Times Tables and Factorials

Now here is a simple idea with which to practise your times tables skills.

If we have six people and six chairs in a row, how many ways can the six people sit on the chairs, with just one person sitting on each chair, of course.

Let’s say the six people are Adam, Beryl, Charlie, Davina, Edward and Fiona.

Only one of the people can sit on the first chair, so let’s assume for a moment that it is Adam.

Now there are five people left to sit on the second chair.

But any of the six people could have sat on the first chair and there would always be five left, any one of which could sit on the second chair. So, that makes 6 x 5 possible combinations of people sitting on the first two chairs. If you don’t believe it, write down their names in a list, like this:

Adam             Beryl
Adam             Charlie
Adam             Davina
Adam             Edward
Adam             Fiona
Beryl              Adam
Beryl              Charlie
Beryl              Davina
Beryl              Edward
Beryl              Fiona
Charlie           Adam
Charlie           Beryl
Charlie           Davina

And so on. You will find 30 pairs altogether.

Let’s now think about the first three chairs. For each pair in the above list, there are four people left, any one of which could sit on the third chair, so the number of combinations for the first three chairs must be 30 x 4, which is 120. It’s just the times tables over and over again.

Now there are just three people left to sit on the fourth chair, so the number of combinations for the first four chairs is 120 x 3, which is 360.

There are now two people for the fifth chair, which makes 360 x 2 = 720 combinations and whoever is left must sit on the remaining sixth chair, so that person has no choice.

The total number of ways in which six people can sit on six chairs is therefore 720. That is probably rather more than you first imagined.

When we multiply a number by all the all the numbers less than the starting number down to 1, we say we are calculating the factorial of the number. So with our chairs we have calculated the factorial of 6. The factorial of 6 is 6 x 5 x 4 x 3 x 2 x 1 = 720

To show this we use the exclamation mark, so we write ‘Factorial 6’ as ‘6!’

Similarly, we write Factorial 12 as ‘12!’

Well, that seemed simple enough, but can we calculate the factorials of other numbers? Of course we can, but it would be better to work logically, so let us begin with factorial 1, which we write as 1! Pretty obviously, one person can sit on one chair in only one way, so we can safely say that:

1! = 1

Now 2! = 2 x 1 = 2
And 3! = 3 x 2 x 1 = 6
And 4! = 4 x 3 x 2 x 1 = 24

Without using a calculator, how far can you calculate the factorial numbers?

BIG HINT:  There is no need to start at the beginning every time. Simply multiply the last factorial by the next number.

For example, we know that 6! = 720, so 7! = 6! x 7 = 720 x 7.

And when you have worked that out, you can calculate 8! by multiplying 7! by 8 and so on.

You will see that the answers grow very quickly, so it will probably take you a very long time to get to 20!

Even with a calculator this is a very difficult calculation because you will soon run out of digits in the display.

So, see how far you can get and good luck, but be careful in your working – one mistake and you will have to start again!!!!!!!!!!! – now, there’s a big factorial.

P.S. What do you think 6!! might mean?