Combinations versus permutations

What you will learn

Choose between combinations and permutations and explain the factorial correction.

Prerequisites: Conditional draws

When order matters

Choosing a first and second card from n distinct cards without replacement gives n(n−1) ordered outcomes. For k positions, continue multiplying n(n−1)…(n−k+1). These are permutations of k cards.

If only the collection matters, each k-card set has k! orderings. Dividing by k! converts ordered sequences to unordered sets: C(n,k)=n!/[k!(n−k)!]. The factorial correction is justified by equal multiplicity, not by a rule to divide whenever a number looks too large.

Represent the same event consistently

There are 52×51 ordered two-card deals but C(52,2)=1326 unordered starting hands. A particular unordered pair appears in two ordered ways. The ratio describing its probability must use either both ordered counts or both unordered counts.

Sampling a uniform shuffled deck induces a uniform distribution on fixed-size unordered subsets. That lets us count sets without losing probability information. Grouping by hand category later does not make the categories equally likely.

Common mistakes

• Using n^k when repeated cards are impossible.
• Dividing by k instead of k!.
• Combining ordered favorable outcomes with an unordered total.

Keep these ideas

• Order-sensitive sequences use falling products.
• Unordered fixed-size sets use combinations.
• Match conventions in numerator and denominator.