Quick Concept Summary
Permutation and combination is structured counting. The first decision in every question is whether order matters: if yes, you arrange (permutation, nPr); if no, you select (combination, nCr). Master the factorial formulas and a handful of standard cases — repetition, alike items, "always together," and circular — and most questions fall quickly.
Core Formulas
| Concept | Formula |
|---|---|
| Factorial | n! = n × (n−1) × … × 1, with 0! = 1 |
| Permutation (arrange r of n) | nPr = n! / (n−r)! |
| Combination (select r of n) | nCr = n! / (r!(n−r)!) |
| Symmetry | nCr = nC(n−r) |
| With repetition (arrange) | nʳ |
| n items, p alike | n! / p! |
| Circular arrangement | (n−1)! |
Order Matters? — The Key Question
- Arrange / rank / form numbers or words → permutation (nPr).
- Choose / select a group / committee → combination (nCr).
- *Select and arrange* → combine: choose with nCr, then arrange with factorials.
Standard Cases
- Always together: bind the group as one unit → arrange units, then within the group.
- Never together: total − (always together).
- Alike items: divide by the factorial of each repeated group (e.g., LEVEL → 5!/(2!·2!)).
- At least / at most: add the valid cases (often using nCr sums).
Worked Examples
Example 1: In how many ways can 5 people sit in a row? 5! = 120.
Example 2: How many 3-digit numbers can be formed from 1–7 with no repetition? 7P3 = 7 × 6 × 5 = 210.
Example 3: A committee of 3 is chosen from 8 people. How many ways? 8C3 = (8 × 7 × 6)/(3 × 2 × 1) = 56.
Example 4: How many arrangements of the letters of "LEVEL"? 5!/(2!·2!) = 120/4 = 30 (L and E each repeat twice).
Example 5: In how many ways can 6 people sit around a round table? (6 − 1)! = 5! = 120.
Common Mistakes
- Using nPr when order does not matter (or nCr when it does).
- Forgetting to divide by the factorial of repeated/alike items.
- Using n! instead of (n−1)! for circular arrangements.
Exam Tips
- First question always: does order matter? That picks nPr vs nCr.
- Remember nCr = nC(n−r) to simplify large selections (e.g., 50C48 = 50C2).
- For "always together," bundle first; for "never together," subtract from the total.