The total number of ways to arrange n distinct objects in a row is n!, and the total number of ways to choose and arrange r objects is nPr.
Factorial and Permutation Formulas
n! = n×(n−1)×…×1、 nPr = n!/(n−r)!
Example ①
Number of ways to arrange 5 people in a row: 5! = 120 ways
Choose a president, vice president, and secretary from 10 people: 10P3 = 10×9×8 = 720 ways
Number of ways to arrange 5 people in a row: 5! = 120 ways
Choose a president, vice president, and secretary from 10 people: 10P3 = 10×9×8 = 720 ways
Permutations with Conditions
Example ② (Adjacent)
Arrangements of 5 people in which A and B are adjacent: Treat AB as one block, giving 4 blocks → 4!×2 = 48 ways
Arrangements of 5 people in which A and B are adjacent: Treat AB as one block, giving 4 blocks → 4!×2 = 48 ways
Key Point
- Adjacent → treat them as one, then multiply by the internal arrangements
- Not adjacent → total − adjacent
Practice Problems
- How many ways are there to arrange 6 people in a row?
- How many ways are there to choose and arrange 3 people from 8 (8P3)?
- How many arrangements of 5 people have A and B not adjacent?
Answer
- 720 ways
- 336 ways
- 120 − 48 = 72 ways