Solve arrangement and selection problems in context using permutations or combinations

What is a Permutation?

A permutation is an arrangement of objects where the order matters. Think of lining up your favourite emojis in a row – the order changes the picture! 😄➡️😃

Formula for arranging all \(n\) objects: $$ P(n) = n! $$

Formula for arranging \(r\) objects out of \(n\): $$ P(n,r) = \frac{n!}{(n-r)!} $$

What is a Combination?

A combination is a selection of objects where the order does NOT matter. Imagine picking a team of 3 friends from a group of 5 – who you pick matters, but the order you name them does not. 🤝

Formula for choosing \(r\) objects out of \(n\): $$ C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} $$

When to Use Which?

• Permutation: When the order of the items matters (e.g., arranging books on a shelf).
• Combination: When the order does not matter (e.g., choosing a hand of cards).

Quick check: Ask yourself, “If I switch two items, does the situation change?” If yes → permutation; if no → combination.

Examples & Practice

Example 1 – Permutation: How many ways can 4 different books be arranged on a shelf?

$$ P(4) = 4! = 24 \text{ ways} $$

Example 2 – Combination: How many ways can you choose 2 ice‑cream flavours from 5 available flavours?

$$ C(5,2) = \binom{5}{2} = \frac{5!}{2!3!} = 10 \text{ ways} $$

Practice Question: In how many ways can 3 students be seated in a row of 5 chairs?

1. Choose 3 chairs out of 5: \( C(5,3) = 10 \) ways.
2. Arrange the 3 students in those chairs: \( P(3) = 3! = 6 \) ways.
3. Total ways: \( 10 \times 6 = 60 \).

Exam Tips

• Always read the question carefully to decide if order matters.
• Use the factorial shortcut: \( n! = n \times (n-1) \times \dots \times 1 \).
• When you see “arrange” → permutation; when you see “choose” or “select” → combination.
• Check your answer by verifying dimensions: permutations give larger numbers than combinations for the same \(n\) and \(r\).
• Practice with real‑life scenarios (e.g., seating, team selection) to build intuition.