Permutations vs. Combinations

The fundamental distinction in counting: permutations count ordered selections; combinations count unordered selections. If you select 3 letters from {A, B, C}: Permutations = 6 (ABC, ACB, BAC, BCA, CAB, CBA). Combinations = 1 ({A, B, C}). The ratio between permutations and combinations is always r! — the number of ways to arrange the selected items.

Permutation Formula: P(n, r) = n! / (n−r)!

Read as "n permute r." Represents the number of ways to arrange r items chosen from n distinct items where order matters. Example: How many 3-digit PINs from digits 1–9 with no repetition? P(9,3) = 9!/6! = 504.

Combination Formula: C(n, r) = n! / (r! × (n−r)!)

Read as "n choose r." The binomial coefficient appears throughout mathematics. Pascal's Triangle: each entry is the sum of the two above, and every entry is C(n,r). The Binomial Theorem: (a+b)ⁿ = Σ C(n,k) × aᵏ × b^(n-k). These appear in probability, calculus, combinatorics, and algebra throughout secondary and university mathematics.

Real-World Examples

Lottery odds: In a 6/49 lottery, C(49,6) = 13,983,816 possible combinations — odds of winning ≈ 1 in 14 million. Committee selection: Choosing a 5-person committee from 20 candidates = C(20,5) = 15,504 ways. Race outcomes: Number of possible 1st/2nd/3rd place finishes in a 10-horse race = P(10,3) = 720 ways. Password security: A 4-digit numeric PIN = 10⁴ = 10,000 (with repetition, not permutation since digits can repeat).