Factorials: The Foundation

Factorial

For a positive integer $n$ , the factorial is defined as:
$n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1$
By convention, $0! = 1$ .

Factorials arise naturally from counting. Suppose you want to arrange $n$ distinct objects in a row. For the first position you have $n$ choices, for the second you have $n - 1$ , and so on — giving $n \times (n - 1) \times \dots \times 1 = n!$ total arrangements.

Example

How many ways can 5 books be arranged on a shelf?

There are $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ arrangements.

Exam Tip

Your GDC can compute factorials directly. On most Casio/TI models, find $n!$ under the probability or math menu. Always verify by hand for small values to build intuition.

Permutations: Ordered Arrangements

Permutation

A permutation is an arrangement of objects in which order matters. The number of ways to arrange $r$ objects chosen from $n$ distinct objects is:
$^{n} P_{r} = \frac{n !}{( n - r )!}$

The logic: you fill $r$ positions one at a time. The first position has $n$ choices, the second has $n - 1$ , …, the $r$ -th has $n - r + 1$ . Multiplying these and rewriting using factorials gives the formula above.

Example

You have 8 plushies and need to arrange 3 of them on a shelf. How many arrangements are possible?

Order matters (the plushie in position 1 is distinct from position 2), so we use permutations:
$^{8} P_{3} = \frac{8 !}{( 8 - 3 )!} = \frac{8 !}{5 !} = 8 \times 7 \times 6 = 336$

There are 336 possible shelf arrangements.

Note

There are several equivalent notations for permutations: $^{n} P_{r}$ , $P (n, r)$ , $P_{r}^{n}$ , $n P r$ — all mean exactly the same thing. The IB formula booklet uses $^{n} P_{r}$ .

Warning

Don't confuse permutations with factorials. $n!$ arranges all $n$ objects; $^{n} P_{r}$ arranges only $r$ of them. In fact, $n! =^{n} P_{n}$ .

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AHL 1.10—Perms and combs, binomial with negative and fractional indices

Factorials: The Foundation

Permutations: Ordered Arrangements

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