Permutation and Combination Calculator
What is a Permutation?
A permutation is an arrangement of objects where order matters. The formula for permutation is: \[P(n,r) = \frac{n!}{(n-r)!}\] where n is the total number of items and r is the number of items being arranged. For example, the number of ways to arrange 3 books out of 5 books is P(5,3) = 60.
What is a Combination?
A combination is a selection of objects where order doesn't matter. The formula for combination is: \[C(n,r) = \frac{n!}{r!(n-r)!}\] where n is the total number of items and r is the number of items being selected. For example, the number of ways to select 3 books out of 5 books is C(5,3) = 10.
Examples and Applications
1. Team Selection: To select a team of 3 players from 11 players:
- If positions matter (specific roles): P(11,3) = 990
- If positions don't matter: C(11,3) = 165
2. Password Generation: Number of 3-digit passwords using digits 0-9:
- With repetition allowed: 10³ = 1000
- Without repetition: P(10,3) = 720
3. Committee Formation: Ways to form a committee of 4 from 20 people:
- C(20,4) = 4845