Systematic Listing & Counting Strategies

What is Systematic Listing?

Systematic listing means writing down every possible outcome in a logical order. The goal is to ensure you don't miss any and don't double count.

Common strategies include:

Alphabetical Order (A, B, C...)
Numerical Order (1, 2, 3...)
Fixing the First Item: Keep the first item the same until you exhaust all options, then move to the next.

1. Systematic Listing Example

Question: You have three cards: 1, 2, and 3. List all the possible 2-digit numbers you can make.

Strategy: Fix the first digit and change the second.

1 and 1 → 11

1 and 2 → 12

1 and 3 → 13

2 and 1 → 21

2 and 2 → 22

2 and 3 → 23

3 and 1 → 31

3 and 2 → 32

3 and 3 → 33

Total outcomes: 9

2. The Product Rule for Counting

Sometimes listing takes too long. If we only need to know how many outcomes there are (not what they are), we use multiplication.

The Rule: If there are $m$ ways to do Task A, and $n$ ways to do Task B, then there are: $$ m \times n $$ ways to do both together.

Example: The Restaurant Menu
A menu has 4 Starters and 5 Main Courses.
How many different 2-course meals can you order?

Calculation: $4 \times 5 = 20$ different meals.

Example: PIN Codes
A 4-digit PIN code uses digits 0-9. How many codes are possible?
Digit 1: 10 options
Digit 2: 10 options
Digit 3: 10 options
Digit 4: 10 options

Calculation: $10 \times 10 \times 10 \times 10 = 10,000$ codes.

Practice Questions & Solutions

Try these yourself before clicking to see the answer!

Question 1: The Coin and Dice (Listing)

Question: A fair coin is flipped (Heads/Tails) and a normal 6-sided dice is rolled. List all possible outcomes.

Solution: We will list systematically by fixing the Coin result first.

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Total: 12 Outcomes

Question 2: The Wardrobe (Product Rule)

Question: Sarah has 3 pairs of jeans, 6 t-shirts, and 2 pairs of sneakers. How many different outfits (Jeans + Shirt + Shoes) can she wear?

Solution: Use the product rule.

$$ 3 \times 6 \times 2 $$ $$ 18 \times 2 = 36 $$ Answer: 36 Outfits

Question 3: Restricted Counting (3-Digit Code)

Question: A 3-digit code is made using the digits 1, 2, 3, 4, 5. The code must be odd. How many combinations are possible?

Solution: Draw 3 boxes: [ ][ ][ ]

Box 3 (Restriction): To be odd, the last number must be 1, 3, or 5. (3 options).

Box 1: Can be any of the 5 digits. (5 options).

Box 2: Can be any of the 5 digits. (5 options).

Calculation: $5 \times 5 \times 3$

Answer: 75 Combinations

Question 4: Arranging People (Permutations)

Question: 4 friends (A, B, C, D) want to sit in a row of 4 chairs. How many ways can they sit?

Solution: Think about filling the chairs one by one.

Chair 1: 4 people available.
Chair 2: 3 people left.
Chair 3: 2 people left.
Chair 4: 1 person left.

$$ 4 \times 3 \times 2 \times 1 $$ Answer: 24 Ways

Question 5: Harder Logic (Combinations)

Question: How many multiples of 5 are there between 1 and 100 that contain the digit '2'?

Solution: Systematic Listing is best here.

1. Multiples of 5 must end in 0 or 5.

2. Let's list those ending in 0 first: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.
(Only '20' contains a 2).

3. Let's list those ending in 5: 15, 25, 35, 45, 55, 65, 75, 85, 95.
(Only '25' contains a 2).

Are there any others? Wait! What about numbers starting with 2?

Check the 20s: 20 (already found), 25 (already found).

Answer: Just 2 (20 and 25).