Counting Cases ① Tree Diagrams and Counting Methods
“Counting cases” is a unit in which we accurately count all possible outcomes. Use tree diagrams to avoid overlooking any cases.
What Is a Tree Diagram?
A tree diagram is a diagram that lists all possible outcomes like the branches of a tree. When dealing with small numbers, this is the most reliable method.
Example 1
When choosing and arranging 2 cards from the 3 cards numbered 1, 2, and 3, how many arrangements are possible?
Solution (tree diagram):
1→{12, 13}
2→{21, 23}
3→{31, 32}
Total: 6 arrangements
When choosing and arranging 2 cards from the 3 cards numbered 1, 2, and 3, how many arrangements are possible?
Solution (tree diagram):
1→{12, 13}
2→{21, 23}
3→{31, 32}
Total: 6 arrangements
Multiplication Principle
When independent operations are performed “one after another,” the number of cases is found by multiplication.
Example 2
There are 3 colors of shirts and 4 colors of pants. How many combinations are there in total?
Solution:
3×4＝12 combinations (multiplication principle)
There are 3 colors of shirts and 4 colors of pants. How many combinations are there in total?
Solution:
3×4＝12 combinations (multiplication principle)
Key Points
- First, develop the habit of listing all cases using a tree diagram
- Independent operations → multiplication (multiplication principle)
- Either one or the other → addition (addition principle)
Practice Problems
- From the 4 people A, B, C, and D, how many ways are there to choose one class representative (class chairperson and vice-chairperson) for each position?
- When a coin is tossed 3 times, how many possible sequences of heads and tails are there in total?
Answers and Explanations
- Answer: 12 ways
Explanation: There are 4 ways to choose the class chairperson. There are 3 ways to choose the vice-chairperson, excluding the class chairperson. By the multiplication principle: 4×3＝12 ways. - Answer: 8 ways
Explanation: There are 2 possibilities, heads or tails, for each toss. The coin is tossed independently 3 times. By the multiplication principle: 2×2×2＝8 ways.