n! (n factorial) means n x (n-1) x (n-2) x ... x 1. So 5! = 5 x 4 x 3 x 2 x 1 = 120. Special case: 0! = 1 (by definition). Factorials count the total number of ways to arrange n distinct objects in a line.

O-Level Add Maths: Permutations and Combinations

Q: What is the difference between a Permutation and a Combination?

A Permutation is an arrangement where ORDER MATTERS. Picking President then Vice-President from 5 people = 5P2 = 20. A Combination is a selection where ORDER DOES NOT MATTER. Picking any 2 people from 5 for a committee = 5C2 = 10.

Q: How do I handle repeated letters in Permutations?

Divide by the factorials of the repeated items. The word MISSISSIPPI has 11 letters with S repeated 4 times, I repeated 4 times, P repeated 2 times. Total arrangements = 11! / (4! x 4! x 2!) = 34,650.

Q: When do I use multiplication vs addition?

Use multiplication (AND) when events happen sequentially: picking a shirt AND a tie = 5 x 3 = 15 outfits. Use addition (OR) when choosing between alternatives: picking a red shirt OR a blue shirt = 3 + 4 = 7 options.

A blackboard with combinatorial diagrams and tree structures.

How do I instantly know if a question is a Permutation or a Combination?

Ask yourself one question: 'Does the order of selection matter?' If the question involves arranging people in a LINE, assigning specific ROLES (1st place, 2nd place), or creating CODES/PASSWORDS — it is a Permutation (nPr). If the question involves choosing a COMMITTEE, selecting a TEAM, or picking ITEMS from a menu — it is a Combination (nCr), because the order you pick them does not change the final group.

1. Factorials: The Building Block
2. Permutations (Order Matters)
3. Combinations (Order Irrelevant)
4. Handling Restrictions and Conditions

Permutations and Combinations is the topic that separates students who memorize formulas from students who understand logic. The formulas are simple. The hard part is reading the English of the question and deciding which formula to apply. This guide from our Ultimate Add Maths Guide trains your decision-making instinct.

1. Factorials: The Building Block

Before touching permutations or combinations, you must deeply understand factorials. n! counts the total number of ways to arrange n distinct objects in a line.

3! = 3 × 2 × 1 = 6 (6 ways to arrange 3 books on a shelf)

5! = 5 × 4 × 3 × 2 × 1 = 120

0! = 1 (by mathematical convention)

10! = 3,628,800 (factorials explode rapidly)

💡 Tutor's Tip

Calculator Shortcut: Your Casio fx-991 has a factorial button. Type the number, then press SHIFT → x^(-1) (which shows x!). This is infinitely faster than multiplying manually, especially for large factorials like 12!.

2. Permutations (Order Matters)

A permutation is an arrangement. ABC is a different permutation from BAC because the order has changed, even though the same 3 letters are used.

Formula: nPr = n! / (n-r)!

Example: How many 3-digit codes can be made from digits 1-7 (no repeats)?

7P3 = 7! / 4! = 7 × 6 × 5 = 210

Repeated Items: The MISSISSIPPI Problem

If items repeat, divide by the factorial of each repeat count. LEVEL has 5 letters, with L repeated twice and E repeated twice. Arrangements = 5! / (2! × 2!) = 120/4 = 30.

3. Combinations (Order Irrelevant)

A combination is a selection. Picking {Alice, Bob} for a committee is the same as picking {Bob, Alice}. The group is identical regardless of the order you chose them.

Formula: nCr = n! / (r! × (n-r)!)

Example: Choose a team of 3 from 8 players.

8C3 = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56

4. Handling Restrictions and Conditions

The CAIE exam never gives you a clean nPr or nCr question. They always add restrictions like "the president must be a woman" or "the letters must start with a vowel".

The Golden Rule: Fix the Restricted Item First

If 5 people must sit in a row and Person A must sit at the end, place A first (2 choices: left end or right end), then arrange the remaining 4 people freely (4! = 24). Total = 2 × 24 = 48.

📋 From the Desk of Joshua Ramirez

The "At Least One" Shortcut:If a question says "at least one woman must be on the committee", do NOT calculate every possible case (1 woman, 2 women, 3 women...). Instead, calculate: Total unrestricted combinations MINUS All-male combinations. This complementary counting method saves enormous time.

Frequently Asked Questions

What is the difference between a Permutation and a Combination?▼

Permutation = arrangement (order matters). Combination = selection (order irrelevant). Locks have combinations, but they should technically be called permutations since 1-2-3 is different from 3-2-1.

What is a Factorial?▼

n! = n x (n-1) x ... x 1. It counts the total arrangements of n distinct objects. 0! = 1 by convention.

How do I handle repeated letters in Permutations?▼

Divide the total factorial by the product of the factorials of each repeated item.

When do I use multiplication vs addition?▼

Multiplication for sequential AND events. Addition for alternative OR events.

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Order vs Chaos: Cracking Permutations and Combinations