The counting principle
If one choice can be made in m ways and a second, independent choice in n ways, both can be made in m × n ways.
Example. Three-digit numbers with different digits from 1–5: 5 × 4 × 3 = 60.
Permutations: order matters
n distinct things in a row: n!. Five people in a row: 5! = 120.
r of n distinct things in order: ⁿPᵣ = n! / (n − r)!.
Combinations: order doesn't matter
ⁿCᵣ = n! / (r!(n − r)!)
Choosing 3 of 8 people: ⁸C₃ = 56.
Handshakes among 10 people (each pair once): ¹⁰C₂ = 45.
Diagonals of an n-sided polygon: ⁿC₂ − n = n(n − 3)/2. A hexagon has 9.
Committees. Two men from 5 and two women from 4: ⁵C₂ × ⁴C₂ = 10 × 6 = 60.
Repeated letters
Arrangements of n letters where one letter repeats p times and another q times: n! / (p! q!).
LETTER: 6 letters with E twice and T twice: 720/4 = 180.
BANANA: 6 letters with A three times and N twice: 720/(6 × 2) = 60.
Circular arrangements
Rotations of a circle give the same seating, so n people around a table can sit in (n − 1)! ways. Six people: 5! = 120.
If clockwise and anticlockwise count as the same (a necklace), divide by 2.
Keeping items together or apart
Together: treat the group as one block, arrange the blocks, then arrange inside the block. Four boys and three girls in a row with the girls together: 5! × 3! = 720.
Never together: total arrangements − arrangements with them together.
How to decide
Question asks for | Use |
|---|---|
arrangements, orders, rankings, numbers, words | permutations |
teams, committees, groups, selections, handshakes | combinations |
Shortcuts and tricks
The standard way is to apply the nPr and nCr formulas. These patterns come up often enough to remember the result directly.
Pairs: n(n − 1)/2
Example: 12 points, no three in a line. How many straight lines join them?
Standard way: ¹²C₂ = 12!/(2! × 10!) = 66.
Shortcut: n(n − 1)/2 = 12 × 11/2 = 66. The same count gives handshakes and matches in a round robin.
Choose the smaller side: nCr = nC(n − r)
Example: ¹⁰C₈.
Standard way: 10!/(8! × 2!) = 45.
Shortcut: ¹⁰C₈ = ¹⁰C₂ = 10 × 9/2 = 45.
"Never together" = total − together
Example: 5 people in a row, two of them never side by side.
Standard way: arrange the other 3 in 3! = 6 ways, then place the two in 2 of the 4 gaps in order: 4 × 3 = 12. Total 6 × 12 = 72.
Shortcut: all arrangements 5! = 120, minus those with the two together, 4! × 2 = 48: 120 − 48 = 72.
Diagonals: n(n − 3)/2
Example: the diagonals of an octagon.
Standard way: ⁸C₂ = 28 lines join the vertices; 8 of them are sides, so 20 are diagonals.
Shortcut: n(n − 3)/2 = 8 × 5/2 = 20.
Exam tips
ⁿCᵣ = ⁿCₙ₋ᵣ, so ¹⁰C₈ = ¹⁰C₂ = 45.
"At least one" is easiest as total − none.
Check whether repetition of digits is allowed.