Permutation and Combination made clear.
Ask one question first: does order matter? Choosing a team is different from arranging that team in seats. This decision determines the counting method.
Key concepts in Permutation and Combination
Multiplication rule
If a task has successive stages with a choices then b choices, there are ab outcomes when each first choice allows b second choices.
Permutation
Arranging r distinct objects from n without repetition: nPr = n!/(n−r)!.
Combination
Choosing r distinct objects from n: nCr = n!/[r!(n−r)!]. Order is ignored.
Repeated objects
Distinct arrangements of n objects with repeated groups of sizes a and b: n!/(a!b!). Circular arrangements of n distinct people are (n−1)! when rotations are identical but reflections differ.
Select or arrange?
| Task | Does order matter? | Method |
|---|---|---|
| Choose a committee | No | Combination |
| Assign president and secretary | Yes | Permutation |
| Arrange books on a shelf | Yes | Factorial for distinct books |
A reliable solving method
- Decide whether you are selecting or ordering.
- Check if repetition is allowed and whether objects are distinct.
- Apply restrictions before multiplying the choices.
- For a small case, list outcomes to check for double-counting.
Step-by-step solved examples
Choose a 3-person team from 5 people.
- Order within the team is irrelevant.
- Use 5C3 = 5!/(3!2!).
- There are 10 teams.
Make a 3-digit number from 1,2,3,4 without repetition.
- Hundreds place has 4 choices.
- Tens has 3 remaining choices; units has 2.
- Total = 4 × 3 × 2 = 24 numbers.
How many distinct arrangements of LEVEL?
- There are five letters, with L twice and E twice.
- Start with 5! and divide by 2! for each repeated pair.
- 5!/(2!2!) = 30 arrangements.
Permutation and Combination: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Choose 2 people from 6.
Show answer and explanation — question 1
Answer B: 15
6C2 = 6×5/2 = 15.
Arrange 4 different books in a row.
Show answer and explanation — question 2
Answer D: 24
4! = 24.
Two-digit numbers using 1,2,3 with repetition allowed?
Show answer and explanation — question 3
Answer C: 9
Each position has three choices: 3×3 = 9.
How many ways to choose all 5 objects from 5?
Show answer and explanation — question 4
Answer B: 1
There is one complete set: 5C5 = 1.
Distinct arrangements of MAMA?
Show answer and explanation — question 5
Answer B: 6
4!/(2!2!) = 6.
Seat 5 distinct people around a round table, rotations identical.
Show answer and explanation — question 6
Answer A: 24
Fix one person to remove rotation: (5−1)! = 24.
Frequently asked questions
Why is 0! equal to 1?
There is one way to arrange an empty selection, and this convention keeps the counting formulas consistent.
Can I always use nCr for a committee?
Only after accounting for restrictions such as required members or specified group representation.
Decide whether you are selecting or ordering. For a small case, list outcomes to check for double-counting.