Factorials, Permutations, Combinations: When Order Matters and When It Doesn't

Some calculations matter for a single decision; this one matters for years. Three related concepts in counting math. Mix them up and your probability problem turns into nonsense.

Factorial (n!)

n × (n−1) × ... × 1. Number of ways to arrange n distinct items. 5! = 120.

Permutations (nPr)

Ordered selections. nPr = n! / (n−r)!. 'How many ways can I rank 3 winners from 10?'

Combinations (nCr)

Unordered selections. nCr = n! / (r! × (n−r)!). 'How many lottery combinations of 6 from 49?'

Real applications

Lottery odds, password strength, deck-of-cards probability, scheduling.

Pro tips

• When asked 'how many ways', the question whether order matters determines P vs C.
• 0! = 1 by convention — makes formulas work.

Frequently asked questions

What are the odds of winning the lottery? 49 choose 6 = 13,983,816. So 1 in nearly 14 million. Don't quit your job.

Why factorial grows so fast? 100! has 158 digits. Multiplication of n × everything below it explodes.

Bookmark the calculator linked above — you will need it more often than you think.