Counting — The Fundamental Principle of Counting

🎯 Learning Objectives

• Apply the fundamental counting principle (multiplication rule) to count outcomes
• Apply the addition rule for mutually exclusive choices
• Compute factorials and understand their combinatorial meaning
• Distinguish between situations requiring the multiplication rule vs addition rule
• Solve basic counting problems with multiple steps

📋 Prerequisites

• Basic arithmetic: multiplication, addition
• No probability knowledge needed — this is the foundation for probability
• BSMA1001-maths-1: Basic set operations (union, intersection) help visualize counting

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📖 Core Content

9.1 Intuition: Why Counting Matters

Everyday analogy: You're at a restaurant choosing a meal: 3 appetizers × 4 main courses × 2 desserts = 24 possible three-course meals. That's counting! 🔑 Key Insight: Most counting problems break down into: "How many ways can we do step 1?" × "How many ways can we do step 2?" × ...

9.2 The Fundamental Counting Principle (Multiplication Rule)

9.2.1 Intuition

9.2.2 Formal Statement

If task A can be done in m ways and, for each of those ways, task B can be done in n ways, then the two tasks together can be done in m × n ways. This extends to any number of tasks: if there are k steps with $n_1, n_2, ..., n_k$ options each, the total number of outcomes is:

$$n_1 \times n_2 \times \cdots \times n_k$$

9.2.3 Worked Examples

$$4 \times 6 \times 3 = 72 \text{ meals}$$ $$26 \times 26 \times 26 \times 10 \times 10 \times 10 = 26^3 \times 10^3 = 17,576 \times 1,000 = 17,576,000$$

• Step 1 (first digit): 9 choices (1-9)
• Step 2 (second digit): 9 choices (0-9, minus the one used)
• Step 3 (third digit): 8 choices
• Step 4 (fourth digit): 7 choices

$$9 \times 9 \times 8 \times 7 = 4,536 \text{ PINs}$$

9.3 The Addition Rule

9.3.1 Intuition

9.3.2 Formal Statement

If task A can be done in m ways and task B can be done in n ways, and the tasks are mutually exclusive (they can't both happen), then the number of ways to do "A or B" is m + n.

9.3.3 Worked Examples

$$5 + 4 = 9$$

• 4-digit passwords: $10^4 = 10,000$
• 3-letter passwords: $26^3 = 17,576$
• Total: $10,000 + 17,576 = 27,576$

9.4 Combining Multiplication and Addition

• Pants outfits: $3 \times 4 = 12$
• Dress outfits: $2$
• Total: $12 + 2 = 14$ Flowchart for choosing counting rule: (Diagram)

9.5 Factorial Notation

9.5.1 Intuition

$$5 \times 4 \times 3 \times 2 \times 1 = 120$$

9.5.2 Definition

$$n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$

• $0! = 1$ (by convention — there's exactly one way to arrange nothing)
• $1! = 1$

9.5.3 Examples

9.6 Worked Examples

Example 1: Multiple-Step Counting (Easy)

$$3 \times 8 \times 5 = 120 \text{ creations}$$

Example 2: With Restrictions (Medium)

• a) Digits can repeat
• b) Digits cannot repeat
• c) The number must be even and digits cannot repeat Solution: a) With repetition: $9 \times 9 \times 9 \times 9 = 9^4 = 6,561$ b) Without repetition: $9 \times 8 \times 7 \times 6 = 3,024$ c) Even, no repetition:
• The last digit must be even: 2, 4, 6, 8 (4 choices)
• First digit: can't be 0 and can't be the last digit → up to 8 choices (9 digits total - 1 used)
• But we need to be careful — the restrictions interact Let's do it step by step:
• Step 1: Choose last digit (must be even): 4 choices (2, 4, 6, 8)
• Step 2: Choose first digit: can't be 0, can't be the chosen last digit → 9 - 1 = 8 choices (from 1-9 minus the last digit)
• Step 3: Choose second digit: any remaining digit from 0-9 except the two used → 10 - 2 = 8 choices
• Step 4: Choose third digit: 10 - 3 = 7 choices

$$4 \times 8 \times 8 \times 7 = 1,792 \text{ even numbers with distinct digits}$$

Example 3: Hotel Room Numbering (Harder)

$$8 \times 2 \times 20 = 320$$

• Floors 5-8: 4 floors
• $4 \times 2 \times 20 = 160$ c) If room 000 is the lobby, and rooms 100-199 are reserved for staff, how many guest rooms?
• Total: 320
• Staff: $1 \times 2 \times 20 = 40$ (floor 1 rooms)
• Lobby: 1 room (000)
• Guest rooms: $320 - 40 - 1 = 279$

9.7 Edge Cases & Gotchas

When Order Doesn't Matter (Preview)

The "At Least" Problem

• Total outcomes: $6^3 = 216$
• Outcome with no 6s: $5^3 = 125$
• Outcomes with at least one 6: $216 - 125 = 91$

Overlapping Events

$$|A \cup B| = |A| + |B| - |A \cap B|$$

9.8 Why This Matters

• Week 6 (Permutations & Combinations): Extends counting to arrangements and selections
• Week 7 (Probability): P(event) = favorable / total (needs counting)
• Week 11 (Distributions): Binomial coefficient in binomial distribution
• BSMA1004 (Stats 2): Counting in hypothesis testing

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📐 Key Formulas / Concepts

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⚠️ Common Pitfalls

Pitfall 1: Using Multiplication When You Should Add

Pitfall 2: Forgetting Restrictions Interact

Pitfall 3: Not Considering 0 as a Digit

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📝 Practice Questions

Q1: Basic Multiplication Rule

A store sells 5 different t-shirts, 3 different jeans, and 4 different shoes. How many outfits (1 shirt + 1 jeans + 1 shoes) can you make?

<details> <strong>Solution</strong>

$5 \times 3 \times 4 = 60 \text{ outfits}$

$\boxed{60}$

</details> > **Q2: Addition Rule** > > You can travel to work by bus (3 routes), train (2 routes), or bike (1 route). How many total travel options? > > <details> <strong>Solution</strong> > > $3 + 2 + 1 = 6 \text{ routes}$ > > $\boxed{6}$ </details> > **Q3: Combined Rules** > > </strong> > > A pizza place offers: build-your-own (choose 1 crust from 3, 1 sauce from 2, and any number of 8 toppings) OR a specialty pizza (10 pre-made options). How many total choices? > > <details> <strong>Solution</strong> > > **Build-your-own:** For each topping, you choose yes/no → 2 choices per topping $3 \times 2 \times 2^8 = 3 \times 2 \times 256 = 1,536$ > > **Specialty:** 10 choices > > **Total:** $1,536 + 10 = 1,546$ > > $\boxed{1,546 \text{ total pizza choices}}$ </details> > **Q4: Factorial Calculation** > > </strong> > > Compute: a) 6! b) 7!/5! c) 10!/8! > > <details> <strong>Solution</strong> > > a) $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ > > b) $\frac{7!}{5!} = \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 = 42$ > > c) $\frac{10!}{8!} = \frac{10 \times 9 \times 8!}{8!} = 10 \times 9 = 90$ > > $\boxed{720,\ 42,\ 90}$ </details> > **Q5: Digit Restrictions** > > </strong> > > How many 3-digit numbers can be formed using digits 1-9 if: a) Digits can repeat b) Digits cannot repeat c) The number is odd and digits cannot repeat > > <details> <strong>Solution</strong> > > **a) With repetition:** $9 \times 9 \times 9 = 9^3 = 729$ > > **b) No repetition:** $9 \times 8 \times 7 = 504$ > > **c) Odd, no repetition:** > > - Last digit must be odd: 1, 3, 5, 7, 9 (5 choices) > - First digit: any remaining digit from 1-9 except the last → 8 choices > - Middle digit: any remaining from 0-9 except first and last → 10 - 2 = 8 choices > > $5 \times 8 \times 8 = 320$ > > $\boxed{729,\ 504,\ 320}$ </details> > **Q6: Password Problem** > > </strong> > > A password must be 4-6 characters long, using lowercase letters (a-z). How many possible passwords? > > <details> <strong>Solution</strong> > > 4 characters: $26^4 = 456,976$ 5 characters: $26^5 = 11,881,376$ 6 characters: $26^6 = 308,915,776$ > > Total: $456,976 + 11,881,376 + 308,915,776 = 321,254,128$ > > $\boxed{321,254,128 \text{ passwords}}$ </details> > **Q7: Complement Counting** > > </strong> > > You roll a die 4 times. How many outcomes have at least one 6? > > <details> <strong>Solution</strong> > > Total outcomes: $6^4 = 1,296$ Outcomes with no 6: $5^4 = 625$ Outcomes with at least one 6: $1,296 - 625 = 671$ > > $\boxed{671}$ </details> > **Q8: Inclusion-Exclusion** > > </strong> > > In a school, 30 students take Statistics, 25 take Calculus, and 12 take both. How many students take Statistics or Calculus? > > <details> <strong>Solution</strong> > > $|S \cup C| = |S| + |C| - |S \cap C| = 30 + 25 - 12 = 43$ > > $\boxed{43 \text{ students}}$ </details> > **Q9: Real-World Application** > > </strong> > > A clothing company makes shirts in 5 sizes (S, M, L, XL, XXL), 8 colors, and 3 sleeve lengths. How many unique SKUs (stock-keeping units) do they need? If they discontinue XXL, how many SKUs are lost? > > <details> <strong>Solution</strong> > > **Total SKUs:** $5 \times 8 \times 3 = 120$ > > **After removing XXL:** $4 \times 8 \times 3 = 96$ > > **SKUs lost:** $120 - 96 = 24$ > > Alternatively: $1 \times 8 \times 3 = 24$ (the XXL size alone) > > $\boxed{120 \text{ total},\ 24 \text{ lost}}$ </details> > **Q10: Complex Counting** > > </strong> > > How many 5-digit numbers (first digit non-zero) are palindromes (read same forwards and backwards)? > > <details> <strong>Solution</strong> > > A 5-digit palindrome looks like: ABCBA > > - A (first digit): 1-9 → 9 choices > - B (second digit): 0-9 → 10 choices > - C (middle digit): 0-9 → 10 choices > - D (fourth digit) = B (same as second) → 1 choice > - E (fifth digit) = A (same as first) → 1 choice > > $9 \times 10 \times 10 \times 1 \times 1 = 900$ > > $\boxed{900 \text{ palindromic 5-digit numbers}}$ </details> * * * ## 🔗 Cross-References - **Next topic:** [Permutations](/notes/01-foundation-bsma1002-stats-1-week06-09-permutations) — arranging items in order - **After that:** [Combinations](/notes/01-foundation-bsma1002-stats-1-week06-10-combinations) — selecting items without order - **Week 7 (Probability):** Using counting to find probabilities - **Week 11 (Binomial Distribution):** The binomial coefficient ${n \choose k}$ - **BSMA1001-maths-1:** Set theory foundations [Join Discord](https://discord.gg/gE2m4Qrdqv) [Previous**Correlation**](/notes/01-foundation-bsma1002-stats-1-week04-07-correlation)[Next**Permutations**](/notes/01-foundation-bsma1002-stats-1-week06-09-permutations)