Lesson 3 of 11 · 7 min

Factorial notation

NCERT §6.3.2

The sports teacher wants to know how many ways 5 runners can line up at the start. The multiplication principle gives 5 × 4 × 3 × 2 × 1. Products like this come up so often that they have their own short name.

The lesson in notes

In short

n! (n factorial) is the product 1 × 2 × 3 × … × n of the first n natural numbers. So 5! = 120, 6! = 720 and 7! = 5040.

0! is defined to be 1. This keeps the formulas for permutations and combinations true at their end cases.

The key step is n! = n × (n − 1)!, and further n! = n(n − 1)(n − 2)! and so on. It lets large factorials cancel instead of being multiplied out.

Worked example: 8!/6! = 8 × 7 × 6!/6! = 56. And 10!/(8! × 2!) = (10 × 9)/2 = 45.

Worked example: find x if 1/7! + 1/8! = x/9!. Multiply every term by 9!: 9!/7! + 9!/8! = x, so 72 + 9 = x and x = 81.

Factorials do not add like ordinary numbers: 3! + 4! = 6 + 24 = 30, while 7! = 5040. For n = 6 and r = 2, n!/(n − r)! = 720/24 = 30.