Further Mathematics · Lesson 17

Permutations, combinations and probability

WAECNECO

What the syllabus asks

WAEC: Statistics and probability: fundamental counting principle; permutations (linear, circular, with repetition and restrictions); combinations; probability (sample space, addition and multiplication laws, mutually exclusive and independent events, conditional probability); binomial distribution (mean and variance); measures of location and dispersion; rank correlation

NECO: Statistics and probability: fundamental counting principle; permutations (linear, circular, with repetition and restrictions); combinations; probability (sample space, addition and multiplication laws, mutually exclusive and independent events, conditional probability); binomial distribution (mean and variance); measures of location and dispersion; rank correlation

What you will learn

Use the counting principle and factorial notation.
Calculate permutations, including repeated objects, circular arrangements and restrictions.
Calculate combinations and solve selection problems, including "at least" cases.
Find probabilities using the addition and multiplication laws.
Solve with- and without-replacement and conditional probability problems.
Apply the binomial distribution, its mean and variance.
Compute mean, variance, standard deviation and Spearman’s rank correlation.

Key points

Counting principle: multiply the number of ways at each stage; fill restricted positions first.
n! arrangements of n different objects; ⁿPᵣ = n!/(n − r)!.
Repeated objects: n!/(p! q! …); circular: (n − 1)!.
Together: block method; apart: total − together.
ⁿCᵣ = n!/[r!(n − r)!] for selections; "at least one" = total − none.
P(A) = n(A)/n(S); P(A′) = 1 − P(A).
P(A ∪ B) = P(A) + P(B) − P(A ∩ B); independent: P(A ∩ B) = P(A)P(B).
Without replacement: change the numerators and denominators; P(A | B) = P(A ∩ B)/P(B).
Binomial: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ; mean np; variance npq.
σ² = Σx²/n − x̄²; Spearman rₛ = 1 − 6Σd²/[n(n² − 1)].

Start here: The fundamental counting principle

Before formulas, there is one big idea. If a first choice can be made in m ways and, for each of these, a second choice in n ways, then the two choices together can be made in m × n ways. This extends to any number of stages.

Ngozi has 3 school shirts and 4 skirts. Each shirt can go with each skirt, so she has 3 × 4 = 12 outfits. Add 2 pairs of shoes and she has 3 × 4 × 2 = 24 different looks.

Worked example 17.1

A vehicle plate in a game uses 3 letters followed by 2 digits. Letters and digits may repeat. How many plates are possible?

Each letter position: 26 ways; each digit position: 10 ways. We multiply because each position is chosen independently.
26 × 26 × 26 × 10 × 10 = 1,757,600.

1,757,600

Worked example 17.2

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if no digit is repeated? How many of these are even?

No repeats: 5 × 4 × 3 = 60.
Even: the last digit must be 2 or 4 (2 ways). Fill it first because it is restricted. Then the first digit: 4 ways; middle: 3 ways. 2 × 4 × 3 = 24.

60 numbers; 24 even

💡 Exam tip: always fill the restricted positions first, then the free ones.

⚠️ Common mistake: adding instead of multiplying (3 + 4 = 7 outfits). Use addition only when the choices are alternatives ("either a shirt or a dress").

Try it yourself. A restaurant offers 4 starters, 5 main dishes and 3 desserts. How many three-course meals are possible?

Show solution

4 × 5 × 3 = 60.

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Practice questions

1. How many arrangements of the letters of ENUGU are there?

Show worked answer

5!/2! = 60.

2. In how many ways can a captain and vice-captain be chosen from 11 players?

Show worked answer

¹¹P₂ = 110.

3. A team of 4 is chosen from 6 boys and 5 girls. How many have exactly 2 girls?

Show worked answer

⁶C₂ × ⁵C₂ = 15 × 10 = 150.

4. Two dice are thrown. Find P(sum = 7).

Show worked answer

6/36 = ⅙.

5. P(A) = 0.4, P(B) = 0.25, independent. Find P(A ∪ B).

Show worked answer

0.4 + 0.25 − 0.1 = 0.55.

Key words

Factorial n!: n × (n − 1) × … × 2 × 1; 0! = 1.
Permutation: An ordered arrangement of objects.
Combination: An unordered selection of objects.
Sample space: The set of all possible outcomes.
Event: A subset of the sample space.
Equally likely: Outcomes with the same chance of happening.
Mutually exclusive: Events that cannot happen together.
Independent events: Events where one does not affect the other’s probability.
Complement: The event "not A", written A′.
Conditional probability: P(A | B), the probability of A given that B has happened.
With replacement: Each item is returned before the next pick.
Binomial experiment: n independent trials, each with the same success probability p.

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