Counting problems concern the skill of accurately determining “how many possibilities there are.” Learn the principles for counting without duplication or omission.
Addition Rule and Multiplication Rule
- Addition rule: Number of cases in which A or B occurs → m + n cases (when mutually exclusive)
- Multiplication rule: Number of cases in which A and B occur consecutively → m × n cases
Example ①
If there are 3 ways to travel from Point A to Point B and 4 ways to travel from Point B to Point C, how many ways are there to travel A→B→C?
Solution: By the multiplication rule, 3 × 4 = 12 ways
If there are 3 ways to travel from Point A to Point B and 4 ways to travel from Point B to Point C, how many ways are there to travel A→B→C?
Solution: By the multiplication rule, 3 × 4 = 12 ways
Inclusion–Exclusion Principle
|A ∪ B| = |A| + |B| − |A ∩ B|
Example ②
Among the integers from 1〜30, find the number of multiples of 3 or 5.
Solution: 10 multiples of 3 + 6 multiples of 5 − 2 multiples of 15 = 14
Among the integers from 1〜30, find the number of multiples of 3 or 5.
Solution: 10 multiples of 3 + 6 multiples of 5 − 2 multiples of 15 = 14
Key Point
- “Or” → addition rule (watch for overlap)
- “And (consecutively)” → multiplication rule
- Overlapping “or” → inclusion–exclusion principle
Practice Problems
- Flip a coin and roll a die simultaneously. How many outcomes are there?
- Among the integers from 1〜20, find the number of multiples of 2 or 3.
- There are 5 ways from Point A to Point B, 3 ways from Point B to Point C, and 2 direct ways from Point A to Point C. How many ways are there from A→C in total?
Solution and Explanation
- Answer: 2 × 6 = 12 ways
- Answer: 10 multiples of 2 + 6 multiples of 3 − 3 multiples of 6 = 13
- Answer: 5×3 + 2 = 17 ways