Mathematics - Additional (0606) | Permutations and combinations

Permutations and Combinations (Syllabus Topic 11)

Welcome to one of the most practical and fascinating topics in Additional Mathematics! Permutations and Combinations are all about counting possibilities. Whether you are dealing with setting a secure password, calculating lottery odds, or arranging books on a shelf, the principles here tell you exactly how many ways something can happen.

Don't worry if this seems tricky at first. We will break down these complex counting methods into two simple categories: when order matters (Permutations) and when order doesn't matter (Combinations). Let's get counting!

1. The Foundations of Counting: Factorials and the Principle

1.1 The Fundamental Counting Principle

This is the most basic rule. If you have multiple independent choices, you simply multiply the number of options for each choice to find the total number of possibilities.

• Rule: If event A can happen in \(m\) ways and event B can happen in \(n\) ways, then A and B together can happen in \(m \times n\) ways.

Example: Imagine choosing an outfit. If you have 3 different shirts and 4 different pairs of trousers, the total number of outfits you can wear is \(3 \times 4 = 12\).

1.2 The Factorial Notation (\(n!\))

The Factorial of a positive integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). Factorials are used when we arrange all available items.

• Definition: \(n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\)

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

\(4! = 4 \times 3 \times 2 \times 1 = 24\) ways

Special Case (Must Know):
The factorial of zero is defined as one: \(0! = 1\). This might seem strange, but it makes the formulas for permutations and combinations work correctly.

Key Takeaway: Counting possibilities starts with multiplication (Fundamental Principle), and factorials are a shortcut for arranging all items.

---

2. Permutations: Arrangement (Order Matters)

2.1 What is a Permutation?

A Permutation is an arrangement of items where the order or position is crucial.

• Key Word: Arrangement.
• Analogy: Think of a race (1st, 2nd, 3rd place are different arrangements of the same people) or a digital code (123 is different from 321).

We use permutations when we are selecting \(r\) items from a total of \(n\) items, and the arrangement of those \(r\) items matters.

2.2 The Permutation Formula (\(^nP_r\))

The number of permutations of \(n\) distinct items taken \(r\) at a time is given by:

\(^{n}P_{r} = P(n, r) = \frac{n!}{(n-r)!}\)

Where:
\(n\) is the total number of items available.
\(r\) is the number of items being selected/arranged.

2.3 Step-by-Step Example: Permutation

Problem: A club has 8 members. How many ways can they choose a President, a Vice-President, and a Treasurer?

Step 1: Determine if order matters. Yes, being President is different from being Vice-President. This is a Permutation.
Step 2: Identify \(n\) and \(r\). Total members \(n=8\). Positions to fill \(r=3\).
Step 3: Apply the formula.

\(^{8}P_{3} = \frac{8!}{(8-3)!} = \frac{8!}{5!}\) \(^{8}P_{3} = \frac{8 \times 7 \times 6 \times 5!}{5!} = 8 \times 7 \times 6 = 336\)
There are 336 ways to fill the three positions.

---

3. Combinations: Selection (Order Doesn't Matter)

3.1 What is a Combination?

A Combination is a selection of items where the order does not matter.

• Key Word: Selection, Group, Team.
• Analogy: Think of picking 3 flavours of ice cream (Chocolate, Strawberry, Vanilla is the same combination as Vanilla, Strawberry, Chocolate).

We use combinations when we are selecting \(r\) items from a total of \(n\) items, and we are only interested in the group formed, not the sequence in which they were picked.

3.2 The Combination Formula (\(^nC_r\))

The number of combinations of \(n\) distinct items taken \(r\) at a time is given by:

\(^{n}C_{r} = C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Notice how this formula is related to the permutation formula:
\(\text{Combination} = \frac{\text{Permutation}}{r!}\)
We divide by \(r!\) because for every group of \(r\) items selected, there are \(r!\) ways to arrange them internally, and we only want to count that group once.

3.3 Step-by-Step Example: Combination

Problem: A club has 8 members. How many ways can they choose a committee of 3 people?

Step 1: Determine if order matters. No, selecting Alice, Bob, and Carol for a committee is the same as selecting Bob, Carol, and Alice. This is a Combination.
Step 2: Identify \(n\) and \(r\). Total members \(n=8\). People selected \(r=3\).
Step 3: Apply the formula.

\(^{8}C_{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!}\) \(^{8}C_{3} = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{6} = 56\)
There are 56 ways to choose the committee.

---

4. Solving Problems: Knowing the Difference

The biggest challenge in this chapter (11.1) is recognising whether a problem requires permutations or combinations.

4.1 The Order Check

Use this simple test:

Imagine you have selected your \(r\) items. If you switch the order of those items, does the result change?

• If YES, the outcome is different: Use Permutations (Arrangement matters).
  Example: Awarding gold, silver, bronze medals. Changing the order changes who gets which medal.
• If NO, the outcome is the same: Use Combinations (Selection matters).
  Example: Choosing 3 lottery numbers. Picking 1, 5, 10 is the same as picking 10, 5, 1.

4.2 Complex Problems Involving Multiple Steps

Sometimes, a problem requires both multiplication (Fundamental Counting Principle) and a selection (Permutations or Combinations).

Example of Multi-Step Selection (Combination used twice):

A class has 10 boys and 8 girls. How many ways can a team of 3 boys and 2 girls be selected?

Step 1: Select the boys. Order doesn't matter (it's a team).
\(n_B = 10\), \(r_B = 3\).
Ways to choose boys:

\(^{10}C_{3} = \frac{10!}{3!7!} = 120\)

Step 2: Select the girls. Order doesn't matter.
\(n_G = 8\), \(r_G = 2\).
Ways to choose girls:

\(^{8}C_{2} = \frac{8!}{2!6!} = 28\)

Step 3: Combine the selections. Use the Fundamental Counting Principle (multiply the independent results).
Total ways = Ways for boys \(\times\) Ways for girls
Total ways = \(120 \times 28 = 3360\)

4.3 Common Mistakes to Avoid

• Mixing up the Formulae: Always check if you should divide by \(r!\) (Combination) or not (Permutation).
• Forgetting \(n\) and \(r\): \(n\) is always the larger number (total available), \(r\) is the smaller number (what you select).
• Not identifying the 'AND' or 'OR' relationship: If choices happen together (AND), you multiply. (This is the Fundamental Counting Principle).

4.4 Syllabus Safety Check (What NOT to worry about)

The 0606 syllabus (11.3) specifically excludes certain complex types of problems. You do not need to worry about the following:

• Repetition of objects: For example, arranging the letters in the word MISSISSIPPI (where some letters are the same). All your problems will involve distinct (different) items.
• Objects arranged in a circle: Formulas for circular arrangements are not required.
• Problems requiring both P & C combined in a single stage: If you use P, you generally won't need C in the same step, and vice versa. (However, solving multi-step problems by multiplying separate C results, as shown in the 3 boys/2 girls example, is required).

Did you know? Factorials grow incredibly fast! \(69!\) is the largest factorial that standard calculators can handle before the result becomes too large to display. This is why these methods are essential for calculating probabilities in large real-world systems.

Key Takeaway: Always start by asking: "Does the order make a difference?" If yes, Permutation. If no, Combination.

---

End of Notes for Permutations and Combinations