Counting② Permutations and Combinations
Let us understand the difference between “arrangements (permutations)” and “selections (combinations).” The key point is whether or not order matters.
Permutations (Arrangements)
Choose and arrange r items from n items (order matters):
nPr ＝ n×(n-1)×…×(n-r+1)
Example 1
In how many ways can 3 people be selected from 5 people and arranged in a line?
Answer:
5×4×3＝60 ways
In how many ways can 3 people be selected from 5 people and arranged in a line?
Answer:
5×4×3＝60 ways
Combinations (Selections)
Choose r items from n items (order does not matter):
nCr ＝ nPr ÷ r!
Example 2
In how many ways can 3 people be selected from 5 people?
Answer:
5C3 ＝ (5×4×3)÷(3×2×1) ＝ 60÷6 ＝ 10 ways
In how many ways can 3 people be selected from 5 people?
Answer:
5C3 ＝ (5×4×3)÷(3×2×1) ＝ 60÷6 ＝ 10 ways
Key Points
- “Order matters” → permutation (nPr)
- “Only selection matters; order does not” → combination (nCr)
- nCr ＝ nC(n-r) (symmetry)
Practice Problems
- In how many ways can 4 people be selected from 6 people?
- In how many ways can a chairperson, vice-chairperson, and secretary be selected from the five people A, B, C, D, and E?
Answers and Explanations
- Answer: 15 ways
Explanation: 6C4＝6C2 (symmetry) ＝ (6×5)÷(2×1) ＝ 30÷2 ＝ 15 ways. - Answer: 60 ways
Explanation: The positions of chairperson, vice-chairperson, and secretary are distinct, so order matters (permutation). 5P3＝5×4×3＝60 ways.