Probability and Combinatorics

Grade 10 · Math · Study Guide

Master counting techniques and probability rules to solve complex chance problems.

Combinatorics is the mathematics of counting. Two fundamental principles are the multiplication rule (if task A can be done in m ways and task B in n ways, together they can be done in m × n ways) and the addition rule for mutually exclusive choices.

Permutations count ordered arrangements: the number of ways to arrange r items chosen from n distinct items is P(n,r) = n!/(n−r)!. When items may repeat in a circular arrangement or some items are identical, adjusted formulas apply.

Combinations count unordered selections: C(n,r) = n!/[r!(n−r)!], also written as ⁿCᵣ or (n choose r). Pascal's triangle provides a visual representation of these values and reveals the binomial theorem.

Probability measures how likely an event is: P(A) = (favourable outcomes)/(total equally likely outcomes). For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B). For non-exclusive events, P(A or B) = P(A) + P(B) − P(A and B).

Conditional probability is the probability of A given B has occurred: P(A|B) = P(A ∩ B)/P(B). Bayes' theorem extends this to update probabilities with new evidence.

Key terms

Permutation

An ordered arrangement of items; P(n,r) = n!/(n−r)! counts the arrangements of r items from n.

Combination

An unordered selection of items; C(n,r) = n!/[r!(n−r)!] counts the selections of r from n.

Factorial

n! = n × (n−1) × ... × 2 × 1; represents the number of ways to arrange n distinct objects.

Mutually Exclusive Events

Events that cannot occur simultaneously; P(A ∩ B) = 0.

Independent Events

Events where the occurrence of one does not affect the probability of the other; P(A ∩ B) = P(A) × P(B).

Conditional Probability

P(A|B) = P(A ∩ B)/P(B); the probability of A given that B has already occurred.

Flashcards

Multiplication Rule

If event A has m outcomes and event B has n outcomes, together they have m × n outcomes.

Permutation Formula

P(n,r) = n!/(n−r)! — ordered selections of r from n distinct items.

Combination Formula

C(n,r) = n!/[r!(n−r)!] — unordered selections of r from n distinct items.

Addition Rule (mutually exclusive)

P(A or B) = P(A) + P(B) when A and B cannot both happen.

Addition Rule (general)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Multiplication Rule (independent)

P(A and B) = P(A) × P(B) when A and B are independent.

Conditional Probability

P(A|B) = P(A ∩ B)/P(B).

Complementary Events

P(not A) = 1 − P(A); the complement covers all outcomes not in A.

Questions

1. How many ways can 4 students be chosen from a class of 10 to form a committee (order does not matter)?

   1. 210
   2. 5040
   3. 40
   4. 720
   Show answer

   Answer: 210

   C(10,4) = 10!/(4! × 6!) = (10×9×8×7)/(4×3×2×1) = 5040/24 = 210.

2. P(A or B) = P(A) + P(B) is always true for any two events A and B.

   Show answer

   Answer: False

   P(A or B) = P(A) + P(B) only holds for mutually exclusive events. In general, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

3. A bag contains 4 red and 6 blue balls. What is the probability of drawing 2 red balls in succession without replacement?

   1. 2/15
   2. 4/25
   3. 8/45
   4. 1/6
   Show answer

   Answer: 2/15

   P(1st red) = 4/10. Given 1st red, P(2nd red) = 3/9 = 1/3. Combined: (4/10)(3/9) = 12/90 = 2/15.

4. The number of ways to arrange all letters in the word LEVEL is ______.

   Show answer

   Answer: 30

   LEVEL has 5 letters with L repeated 2 times and E repeated 2 times. Arrangements = 5!/(2!×2!) = 120/4 = 30.

5. How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition?

   1. 60
   2. 125
   3. 20
   4. 10
   Show answer

   Answer: 60

   P(5,3) = 5!/(5−3)! = 5!/2! = 60.

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