Factorials

Before your students begin this activity, you may need to discuss with them the information given at the top of the page to ensure that they understand what factorials are and how they are written. In factorials, the exclamation mark (!) signifies that the number should be multiplied by consecutively lower whole numbers. Thus 5! (5 factorial) is 5 x 4 x 3 x 2 x 1.
The activity explores factorials as pure numbers and then applies factorials to a real life situation. However, as an extra challenge for the students, you could reverse the order. You could ask them to begin with the bus queue problem and then discuss with them whether there is some clever mathematical short cut to working out problems of this nature.
Although the students could use a calculator to work out solutions to the questions fairly easily, encourage them to look for patterns that will help them solve the problems quickly, without using a calculator. To detect patterns, they will need to understand how the numbers are working. For example, to work out what 9! is
when 10! is known, the students need to understand that 10! is 10 times greater than 9! (or that 6! is 6 times greater than 5!, and so on). Similarly, in question 4, the students should be able to detect a pattern reasonably easily (3! divided by 1! is the same number as 3 x 2, 4! divided by 2! is the same number as 4 x 3, and so
on) and to use this pattern to predict the solutions to questions 4b and 4c. Ask the students to explain why this pattern works. They might describe it this way:
3! divided by 1! is really

The 1 in the denominator cancels out the 1 in the numerator, so you are left with 3 x 2. So 3! ÷ 1! = 6.
4! divided by 2! is really

5! divided by 3! is really

Such an understanding will enable the students to work out other examples, such as 8! ÷ 6!, which, in short, is 8 x 7 because it is really

This “cancelling out” is the power of the identity principle. The relevant aspect here is the idea that dividing any number by itself results in 1. Take the last example above,

The part now shown in brackets is simply 1. It is not even necessary to work out that the bracketed part of the top is 720 and that the bottom is also 720. Logic indicates that they are the same number, whatever that number might be.

Answers to Activity

1. a. 5! = 5 x 4 x 3 x 2 x 1 = 120
b. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
c. 10! = 10 x 9 x 8 ... = 3 628 800
2. You could divide 3 628 800 (10!) by 10 because 10! = 9! x 10. 3 628 800 ÷ 10 = 362 880
3. 7! = 8! ÷ 8
= 40 320 ÷ 8
= 5 040
4. a. 3! ÷ 1! is the same as 3 x 2, 4! ÷ 2! is the same as 4 x 3, and 5! ÷ 3! is the same as 5 x 4. So the pattern is: multiply the initial factor by the number that is one more than the number shown as the second factorial.
b.–c.
Based on the pattern in 4a, 6! ÷ 4! is the same as 6 x 5, which is 30, and 10! ÷ 8! is the same as 10 x 9, which is 90. You can check your predictions based on the
following:
1! = 1
2! = 2 x 1
3! = 3 x 2 x 1
4! = 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
6! = 6 x 5 x 4 x 3 x 2 x 1
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
So:
3! ÷ 1! = 6 ÷ 1
= 6, which is 3 x 2
4! ÷ 2! = 24 ÷ 2
= 12, which is 4 x 3
5! ÷ 3! = 120 ÷ 6
= 20, which is 5 x 4
6! ÷ 4! = 720 ÷ 24
= 30, which is 6 x 5
10! ÷ 8! = 362 8800 ÷ 40 320
You can also set these out using the factors.
For example,

5. a. i. 2 people → 2 ways: ab, ba (2!)
ii. 3 people → 6 ways: abc, acb, bac, bca, cab, cba (3!)
iii. 4 people → 24 ways: abcd, abdc, ... (4!)
b. The number of ways corresponds to the factorial of the number of people being studied.
c. 3 628 800 ways (10!)