Lesson

Combinatorics

Combinatorics counts arrangements and selections. A permutation $P(n, r)$ counts ordered selections; a combination $C(n, r) = \dfrac{n!}{r!(n-r)!}$ counts unordered selections.

Practice

Q1

Compute $C(5, 2)$.

Show hint

$\frac{5\cdot4}{2}$.

Q2

Compute $P(4, 2)$.

Show hint

Use $P(n,r) = n!/(n-r)!$.

Factorials grow quickly and appear throughout counting. Note the useful identities $C(n, 0) = 1$ and $C(n, n) = 1$.

Quiz

Q1

Compute $C(6, 0)$.

Q2

Compute $5!$ (five factorial).

Show hint

Multiply 5 × 4 × 3 × 2 × 1.

Q3

Compute $C(4, 2)$.

Show hint

It equals (4 × 3) / 2.