Mathematics · Lesson 28

Permutations & combinations

JAMBWAECNECO

What the syllabus asks

JAMB: Permutations and combinations: simple cases, the counting principle, factorial notation, nPr and nCr, arrangements with repeated objects, applications to selection problems and probability

WAEC: counting and arrangement problems used in probability (outcomes, selections); Further Mathematics develops the topic fully

NECO: counting and arrangement problems used in probability (outcomes, selections); Further Mathematics develops the topic fully

What you will learn

Use the counting principle to find the number of outcomes of successive choices.
Evaluate and simplify factorial expressions.
Distinguish between situations where order matters (permutations) and does not (combinations).
Use nPr and nCr to count arrangements and selections.
Count arrangements of words with repeated letters and arrangements with restrictions.
Solve committee and selection problems, including “at least” cases.
Solve equations involving nPr and nCr and apply counting to probability.

Key points

Successive choices multiply (counting principle); alternative choices add.
n! = n(n − 1)…1, 0! = 1; simplify factorial fractions by cancelling.
Order matters → permutation: nPr = n!/(n − r)!.
Order does not matter → combination: nCr = n!/(r!(n − r)!); nCr = nC(n − r).
nPr = r! × nCr.
Repeated objects: n!/(p! q! …).
Circular arrangements: (n − 1)!.
Restrictions first; “together” means glue then multiply by internal arrangements; “not together” = total − together.
Committees from groups: multiply combinations; “at least one” = total − none.
Probability = favourable count ÷ total count.

Start here: The counting principle

Chiamaka is dressing for her cousin’s wedding in Owerri. She has 3 blouses and 4 wrappers. How many different outfits can she make? For each blouse there are 4 wrappers, so there are 4 + 4 + 4 = 3 × 4 = 12 outfits. This simple idea is the foundation of the whole topic.

Counting principle: if a first choice can be made in m ways and, after it, a second choice in n ways, then the two choices together can be made in m × n ways. It extends to any number of stages.

A tree diagram shows why: 3 first branches, each splitting into 4, gives 12 end points. But drawing trees becomes impossible with large numbers, so we multiply.

Worked example 28.1

A canteen offers 4 main dishes (jollof rice, fried rice, amala, pounded yam), 3 soups or sauces and 2 drinks. How many different meals of one of each are possible?

There are three stages: main dish, sauce, drink.
Multiply: 4 × 3 × 2 = 24.

24 meals

Worked example 28.2

How many three-digit numbers can be formed from the digits 1, 2, 3, 4, 5 if (a) repetition is allowed, (b) no digit may be repeated?

(a) Each of the three places can be any of the 5 digits: 5 × 5 × 5 = 125.
(b) The hundreds digit has 5 choices; the tens digit then has 4 (one is used); the units digit has 3. So 5 × 4 × 3 = 60.

(a) 125, (b) 60

Worked example 28.3 (restriction first)

How many of the numbers in 28.2(b) are even?

Deal with the restricted place first. An even number must end in 2 or 4: 2 choices for the units.
Then the hundreds digit has 4 choices left and the tens digit 3.
Total = 2 × 4 × 3 = 24.

24 even numbers

⚠️ Common mistake: adding choices instead of multiplying when they happen together. Add only when you choose one option OR another (e.g. travel by bus (3 routes) or by train (2 routes): 3 + 2 = 5 ways).

Try it yourself. A password is one letter (A to Z) followed by two digits (0 to 9), repetition allowed. How many passwords are possible?

Show solution

26 × 10 × 10 = 2,600.

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Practice questions

1. Evaluate 6P2 and 6C2.

Show worked answer

6P2 = 30; 6C2 = 15.

2. In how many ways can a chairman, secretary and treasurer be chosen from 7 people?

Show worked answer

7P3 = 7 × 6 × 5 = 210.

3. How many arrangements are there of the letters of KANO? Of OYO?

Show worked answer

4! = 24; 3!/2! = 3.

4. How many arrangements are there of the letters of BENUE?

Show worked answer

5!/2! = 60.

5. A committee of 2 men and 3 women is chosen from 7 men and 5 women. How many committees are possible?

Show worked answer

21 × 10 = 210.

Key words

Outcome: One possible result of an action or experiment.
Counting (multiplication) principle: The rule that numbers of choices at successive stages are multiplied.
Factorial (n!): The product of all whole numbers from n down to 1.
Permutation: An ordered arrangement of objects.
Combination: An unordered selection of objects.
nPr: The number of ordered arrangements of r objects chosen from n different objects.
nCr: The number of selections of r objects from n different objects; also written ⁿCᵣ or (n r).
Distinct objects: Objects that can all be told apart.
Identical (alike) objects: Objects that cannot be told apart, such as two letters A.
Restriction: A condition placed on an arrangement, such as “must start with a vowel”.
Circular permutation: An arrangement around a circle, where rotations are counted as the same.
Complement: Everything not in the event; used for “at least one” counts.

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