Probability and Combinatorics
Grade 11 · Math · Study Guide
Count arrangements with permutations and combinations, then calculate probabilities of single and compound events.
A permutation P(n,r) = n!/(n-r)! counts ordered arrangements of r items from n. A combination C(n,r) = n!/(r!(n-r)!) counts unordered selections. Use combinations when order does not matter.
Probability of event A: P(A) = (favorable outcomes)/(total outcomes) for equally likely outcomes. The complement rule: P(A') = 1 - P(A).
For compound events: P(A or B) = P(A) + P(B) - P(A and B). For independent events: P(A and B) = P(A) * P(B). Conditional probability: P(A|B) = P(A and B) / P(B).
| Formula | Expression | When to Use |
|---|---|---|
| Permutation | P(n,r) = n!/(n-r)! | Order matters |
| Combination | C(n,r) = n!/(r!(n-r)!) | Order does not matter |
| Fundamental Counting Principle | m x n x ... | Independent sequential choices |
| Rule | Formula |
|---|---|
| Complement | P(A') = 1 - P(A) |
| Addition (general) | P(A or B) = P(A) + P(B) - P(A and B) |
| Multiplication (independent) | P(A and B) = P(A) * P(B) |
| Conditional | P(A|B) = P(A and B) / P(B) |
Key terms
- Permutation
- An ordered arrangement of items; P(n,r) = n!/(n-r)!.
- Combination
- An unordered selection of items; C(n,r) = n!/(r!(n-r)!).
- Complement
- All outcomes NOT in event A; P(A') = 1 - P(A).
- Conditional probability
- P(A|B) = probability of A given B occurred = P(A and B)/P(B).
Practice questions
How many ways can you arrange 4 books on a shelf chosen from 7 (order matters)?
- 28
- 210
- 840
- 35
Show answer
P(7,4) = 7!/(7-4)! = 7*6*5*4 = 840.
How many 3-person committees can be formed from 8 people?
- 24
- 336
- 56
- 512
Show answer
C(8,3) = 8!/(3!*5!) = (8*7*6)/(3*2*1) = 336/6 = 56.
A bag has 3 red and 7 blue marbles. What is the probability of picking a red marble?
Show answer
P(red) = 3/(3+7) = 3/10 = 0.3.
P(A) = 0.4 and P(B) = 0.5, and A and B are independent. What is P(A and B)?
- 0.9
- 0.1
- 0.2
- 0.45
Show answer
P(A and B) = P(A)*P(B) = 0.4 * 0.5 = 0.2.
What is the probability of NOT rolling a 6 on a standard die?
Show answer
P(not 6) = 1 - P(6) = 1 - 1/6 = 5/6.
Fun facts
- The number of ways to arrange a standard deck of 52 cards is 52! — a number so astronomically large that every shuffle in history has almost certainly been unique.
- Pascal's Triangle contains the combination values C(n,r) as its entries, directly connecting combinatorics to binomial expansion.
Get a personalized course on Probability and Combinatorics built around your child.