A permutation is an arrangement (order matters); a combination is a selection (order does not matter).
1Factorials
n! = n × (n − 1) × … × 1, and 0! = 1
2Counting
ⁿPᵣ = n!/(n − r)!
ⁿCᵣ = n!/(r!(n − r)!), and ⁿCᵣ = ⁿCₙ₋ᵣ
Arrangements of n items with p alike of one kind, q alike of another: n!/(p! q!)
Circular arrangements of n distinct items: (n − 1)!
3Handy results
Handshakes among n people: ⁿC₂
Diagonals of an n-sided polygon: n(n − 3)/2
Items that must be together: treat them as one unit, then multiply by their internal arrangements