Systematic Listing and Counting
This topic covers systematic listing, a method to calculate the total number of possible outcomes for a sequence of events. It's a foundational skill for probability and combinatorics, tested by finding the product of the number of options available at each stage.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The core principle is the product rule: if you have 'm' ways to do one thing and 'n' ways to do another, you have m × n ways to do both in sequence.
- This rule extends to any number of sequential tasks. For example, three tasks with m, n, and p options give m × n × p total combinations.
- Pay close attention to whether choices can be repeated. If an item cannot be reused (e.g., 'digits must be unique'), the number of options decreases at each subsequent step.
- If a problem presents alternative scenarios (e.g., 'a code can have 3 OR 4 digits'), calculate the possibilities for each scenario separately and then add the results together.
Formulae
Total Outcomes = m × n × p × ⋯ To find the total number of combinations for a sequence of events, where 'm', 'n', 'p' are the number of options for each consecutive event.
Definitions
- Systematic Listing
- An organised method of enumerating all possible outcomes of a sequence of choices or events.
- Product Rule for Counting
- The principle that the total number of ways to perform a series of tasks is the product of the number of ways to perform each individual task.
Worked example
A security PIN has 4 characters. The first character must be a vowel (A, E, I, O, U). The second must be a prime number less than 10. The third and fourth characters must be different digits from 0-9, and neither can be the same as the second character. How many different PINs are possible?
- 1
Step 1:
Identify the options for the first character.
There are 5 vowels (A, E, I, O, U), so there are 5 choices.
- 2
Step 2:
Identify the options for the second character.
The prime numbers less than 10 are 2, 3, 5, 7.
This gives 4 choices.
- 3
Step 3:
Identify the options for the third character.
There are 10 digits (0-9), but it cannot be the same as the digit used in the second character.
This leaves 10 - 1 = 9 choices.
- 4
Step 4:
Identify the options for the fourth character.
It must be different from the third character and the second character.
We have already used two distinct digits, leaving 10 - 2 = 8 choices.
- 5
Step 5:
Multiply the number of choices for each position:
5 × 4 × 9 × 8.
- 6
Step 6:
Calculate the final product:
20 × 72 = 1440
Answer: 1440
Common mistakes
- ×Forgetting to reduce the number of options for subsequent choices when items cannot be repeated (e.g., using 10 × 10 instead of 10 × 9 for two unique digits).
- ×Incorrectly adding choices instead of multiplying for a sequence of events. Addition is for 'OR' scenarios, multiplication is for 'AND' scenarios.
- ×Miscounting the number of initial options, for example, by forgetting that 0 is a digit or misidentifying primes or vowels.
No-calculator tips
- ✓When multiplying a list of numbers, look for pairs that make multiples of 10 first. For example, in 5 × 7 × 4, calculate 5 × 4 = 20 first, then 20 × 7 = 140.
- ✓Break down two-digit multiplication. To calculate 15 × 18, you could do (15 × 10) + (15 × 8) = 150 + 120 = 270.
- ✓For products like 26 × 25, it can be easier to think of it as 26 × 100 / 4, which is 2600 / 4 = 650.