60-second revision
- AND means multiply the counts; OR (mutually exclusive) means add them.
- nPr = n!/(n − r)! counts arrangements; nCr = n!/(r!(n − r)!) counts selections.
- Arrangements with repeats: n!/(p! q! ...); around a circle: (n − 1)!.
- Items that must stay together: treat them as one unit, then multiply by their internal arrangements.
- P(event) = favourable ÷ total; P(not E) = 1 − P(E).
- P(A or B) = P(A) + P(B) − P(A and B).
- Two dice give 36 outcomes; a sum of 7 is the most likely (6 ways).
Two counting principles
- Multiplication (AND): if one step can be done in m ways and the next in n ways, both together can be done in m × n ways. 3 shirts and 4 trousers give 12 outfits.
- Addition (OR): if a task can be done in m ways OR in n separate ways, there are m + n ways in all.
n! (n factorial) = n × (n − 1) × ... × 1, and 0! = 1. Useful values: 4! = 24, 5! = 120, 6! = 720, 7! = 5040.
Permutations and combinations
Use permutations when order matters (arrangements, ranks, seats, numbers). Use combinations when it does not (teams, committees, handshakes, choosing balls).
| Situation | Count |
|---|---|
| Arrange n different things in a row | n! |
| Arrange r out of n different things | nPr = n!/(n − r)! |
| Choose r out of n | nCr = n!/(r!(n − r)!), and nCr = nC(n − r) |
| Arrange n things with p alike and q alike | n!/(p! × q!) |
| Seat n people around a round table | (n − 1)! |
| Necklace or garland of n different beads | (n − 1)!/2 |
| Handshakes among n people | nC2 |
| Diagonals of an n-sided polygon | nC2 − n |
Worked example: MANGO has 5 different letters, so 5! = 120 arrangements. BANANA has A three times and N twice: 6!/(3! × 2!) = 60.
Worked example: ORANGE with all vowels (O, A, E) together. Treat the vowels as one block: 4 units give 4! = 24, and the block arranges itself in 3! = 6 ways. Total 144.
Worked example: a team of 2 men and 2 women from 5 men and 4 women can be chosen in C(5,2) × C(4,2) = 10 × 6 = 60 ways.
Selections with conditions
- A particular person must be included: fix them in, choose the rest, C(n − 1, r − 1). From 8 players, teams of 5 that include the captain = C(7,4) = 35.
- A particular person must be left out: C(n − 1, r). Teams of 5 without the captain = C(7,5) = 21. Check: 35 + 21 = 56 = C(8,5).
- A word must start with a given letter: fix it and arrange the rest. Arrangements of PLANET starting with P = 5! = 120.
- Two people must NOT sit together in a row of n: total n! minus the arrangements where they sit together, 2 × (n − 1)!.
Probability basics
Probability = number of favourable outcomes ÷ total number of equally likely outcomes. It always lies between 0 and 1.
- P(not E) = 1 − P(E).
- P(A or B) = P(A) + P(B) − P(A and B); drop the last term only if A and B cannot happen together.
- Independent events: P(A and B) = P(A) × P(B).
| Experiment | Total outcomes | Remember |
|---|---|---|
| One coin / n coins | 2 / 2^n | At least one head = 1 − 1/2^n |
| One die / two dice | 6 / 36 | Doubles: 6 of 36 |
| Pack of cards | 52 | 4 suits of 13; 26 red, 26 black; 12 face cards; 4 aces |
| Sum of two dice | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
Worked example: P(sum of two dice = 9) = 4/36 = 1/9. P(at least one head in 4 tosses) = 1 − 1/16 = 15/16.
Worked example: drawing 2 balls from 3 red and 5 blue, P(one of each) = (3 × 5)/C(8,2) = 15/28.
Worked example: P(red card or king) = (26 + 4 − 2)/52 = 28/52 = 7/13, because the two red kings were counted twice.
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