Business Mathematics and Basic Statistics · Ch 8 — Permutations and Combinations

Factorial Notation

Business Mathematics and Basic Statistics (WBCHSE Class 11 Commerce) begins its treatment of counting techniques with the factorial, the building block every permutation and combination formula in this chapter is built from.

What is a factorial?

For a whole number $n$ , the factorial of $n$ , written $n!$ (read " $n$ factorial"), is the product of all whole numbers from $n$ down to $1$ :

n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1, \qquad n \in \mathbb{W},\ n \ge 1

So $4! = 4 \times 3 \times 2 \times 1 = 24$ , and $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ .

Note

The axiom $0! = 1$

Zero factorial is DEFINED to equal $1$ — this is stated as an axiom (a starting rule we accept), not something derived by "multiplying zero numbers together". It is fixed this way precisely so that the $^{n}P_{r}$ and $^{n}C_{r}$ formulas you meet later in this chapter keep working correctly at their boundary cases, such as $^{n}P_{0} = 1$ and $^{n}C_{0} = 1$ .

A quick reference table

$n$	0	1	2	3	4	5	6	7	8
$n!$	1	1	2	6	24	120	720	5040	40320

Factorials grow very fast — exactly why this WBCHSE Class 11 Commerce Business Mathematics and Basic Statistics syllabus caps every permutations-and-combinations problem at $n \le 8$ : beyond that the ARITHMETIC (never the idea) becomes unwieldy for a semester-exam setting.

A useful recursive idea

Every factorial can be written in terms of the one just before it:

n! = n \times (n-1)!

For example, $5! = 5 \times 4! = 5 \times 24 = 120$ . This trick is usually the fastest way to simplify a ratio of two factorials, such as $\dfrac{7!}{5!}$ , without multiplying everything out first:

\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42

This cancellation habit is exactly what makes the $^{n}P_{r}$ and $^{n}C_{r}$ formulas ahead quick to evaluate by hand.

Definition 1Factorial (n!)

For a whole number $n \ge 1$ , $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$ . By convention — an axiom, not a derived fact — $0! = 1$ .

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