Counting principle and factorial
The multiplication principle: if one event can occur in m ways and another in n ways, both together occur in m × n ways.
The factorial n! = n(n−1)(n−2)…(2)(1), and by definition 0! = 1. For example 5! = 120.
Permutations
A permutation is an ordered arrangement. Arranging all n distinct objects gives n! arrangements. Arranging r chosen from n gives ⁿPr.
Key formula
ⁿPr = n!/(n − r)!. Arrangements of n objects with repeats (p alike of one kind, q of another): n!/(p! q! …).
Worked example
Worked example
How many 3-letter arrangements can be made from 5 distinct letters? ⁵P₃ = 5!/(5−3)! = 5!/2! = 120/2 = 60.
With repeated letters: the word "BOOK" has 4 letters with O repeated twice, so the number of distinct arrangements is 4!/2! = 24/2 = 12.
Remember
- Order matters in a permutation.
- ⁿPn = n! and ⁿP₁ = n.
- Divide by the factorial of each repeated group.