Permutation and Combination

Order matters → permutation. Order does not matter → combination. Ask 'would swapping two picks give a different answer?'

Start with this

How many ways can 3 people stand in a line? Try it: 6. How many ways to pick 2 of them for a team? 3. Arranging and choosing are different questions.

Counting is the engine of probability too. The whole topic: multiply for 'and', add for 'or', use P when order matters and C when it does not.

The counting principle: boxes to fill

Worked example

How many 3-digit numbers can be formed with the digits 1 to 6 if no digit repeats?

1. Hundreds: 6 choices. Tens: 5. Units: 4.
   Why: Each used digit is no longer available.
2. 6 × 5 × 4 = 120.

Answer: 120

Worked example

How many 3-digit numbers with digits from 0 to 9, repetition allowed?

1. Hundreds: 9 choices (no 0 in front). Tens: 10. Units: 10.
   Why: A number cannot start with 0.
2. 900.

Answer: 900

Permutations: order matters

nPr = n! / (n - r)! = n × (n - 1) x ... (r numbers). Arranging all n things: n!.

Identical items: divide by the factorial of each repeat. Arrangements of BANANA = 6! / (3! 2!) = 60.

Items that must stay together: glue them into one block, arrange, then arrange inside the block.

Worked example

In how many ways can the letters of LEADER be arranged?

1. 6 letters, E twice.
   Why: Repeated letters make some arrangements identical.
2. 6! / 2! = 360.

Answer: 360

Worked example

In how many arrangements of the word APPLE do the two Ps come together?

1. Glue PP into one block: A, PP, L, E = 4 items.
   Why: Together means one block.
2. 4! = 24.
   Why: The two Ps are identical, so no inside arrangement.
3. 24.

Answer: 24

Combinations: order does not matter

nCr = n! / (r! (n - r)!) = nPr / r!. Teams, committees, handshakes, groups: choosing, not arranging.

nCr = nC(n - r): choosing 3 to play is the same as choosing 7 to sit out of 10.

Worked example

A committee of 3 men and 2 women is to be chosen from 6 men and 5 women. How many ways?

1. Men: 6C3 = 20. Women: 5C2 = 10.
   Why: Choose each group separately.
2. 20 × 10 = 200.

Answer: 200

Worked example

At least one woman: a team of 3 from 4 men and 3 women.

1. All teams: 7C3 = 35.
2. No women: 4C3 = 4.
   Why: 'At least one' → total minus none.
3. 35 - 4 = 31.

Answer: 31

Circles and restrictions

n people around a table: (n - 1)!. Fix one person; arrange the rest. Necklaces and garlands that can be flipped: (n - 1)!/2.

Two people must NOT sit together: total - (arrangements where they are together).

Worked example

In how many ways can 6 people sit around a round table?

1. Fix one person. Arrange 5 others.
   Why: Rotations of the same seating count once.
2. 5! = 120.

Answer: 120

Worked example

5 people in a row. Two of them, A and B, refuse to sit together. How many ways?

1. All: 5! = 120.
2. A and B together: glue them, 4! x 2! = 48.
   Why: The block can be AB or BA.
3. 120 - 48 = 72.

Answer: 72

Shortcut: Glue the block

Use it when: 'Vowels together'.

1. ORANGE: vowels O, A, E become one block.
2. Arrange 4 things: 4!. Inside the block: 3!.
3. 24 × 6 = 144.

Glue the block: try it

Arrangements of ORANGE with vowels together

1. 144.

Answer: 144

Common mistakes

Your turn medium

4-digit numbers with no repeated digit?

1. 9 × 9 × 8 × 7.

Answer: 4536

Your turn hard

In how many ways can 5 boys and 3 girls sit in a row so that no two girls sit together?

1. Seat boys: 5!. 6 gaps, pick and order 3: 6P3 = 120.
2. 120 × 120.

Answer: 14,400

Your turn hard

A team of 5 is chosen from 7 men and 4 women with at least 2 women. Ways?

1. 2W: 6 × 35 = 210. 3W: 4 × 21 = 84. 4W: 1 × 7 = 7.

Answer: 301

Your turn hard

Arrangements of the letters of MATHEMATICS?

1. 11!/(2! 2! 2!).

Answer: 49,89,600

Recap

• draw boxes and multiply
• choose P or C by asking 'does order matter?'
• handle repeats, blocks, circles and 'not together'

Now practise

13 questions with full solutions.

Open practice Back to the lesson