Chapter 4 – Permutations and Combinations (JEE)
1. Fundamental Principle of Counting
If one task can be done in m ways and another task can be done in n ways,
then the total number of ways to perform both tasks is m × n.
This principle is the backbone of the entire chapter.
Every permutation and combination problem is ultimately based on this idea.
2. Factorial Notation
$n! = n × (n-1) × (n-2) × \dots × 1$
- $0! = 1$
- $1! = 1$
Factorials grow very fast. Always simplify before calculating.
3. What is a Permutation?
Permutation means arrangement.
Here, order matters.
Number of permutations of $n$ different objects taken $r$ at a time:
$$nP_r = \frac{n!}{(n-r)!}$$
4. Permutation of All Objects
Number of arrangements of $n$ distinct objects = $n!$
5. Permutations with Repetition
If repetitions are allowed and each position has $n$ choices,
then number of arrangements for $r$ places is:
$n^r$
6. Permutations of Identical Objects
If among $n$ objects, $p$ are identical of one kind,
$q$ are identical of another kind, then:
$$\frac{n!}{p!q!}$$
Very common in word problems (like letters of a word).
7. Circular Permutations
Arrangements in a circle where clockwise and anticlockwise are different:
$(n-1)!$
If clockwise and anticlockwise are considered same → divide by 2.
8. What is a Combination?
Combination means selection.
Here, order does NOT matter.
Number of combinations:
$$nC_r = \frac{n!}{r!(n-r)!}$$
9. Important Properties of Combinations
$nC_r = nC_{n-r}$
$nC_0 = nC_n = 1$
10. Relation Between Permutation and Combination
$nP_r = nC_r \times r!$
11. Selection with Restrictions
Many JEE questions involve conditions like:
- At least one particular object
- At most one
- Exactly one
Use cases:
- Total ways − restricted ways
- Fix one object and select remaining
12. Distribution of Objects
Important cases:
- Identical objects into distinct boxes
- Distinct objects into identical boxes
This part requires careful case analysis.
13. Combination Problems Using Algebra
Expressions involving $nC_r$ often simplify using identities.
$nC_r + nC_{r-1} = (n+1)C_r$
14. Common JEE Traps
- Confusing permutation with combination
- Forgetting identical objects
- Ignoring circular nature
- Not considering restrictions properly
15. Typical JEE Question Patterns
| Pattern | Approach |
|---|---|
| Arrangement of letters | Use identical permutations |
| Selection of students | Use combinations |
| Circular seating | Use circular permutation |
| Binary strings | Use $n^r$ or combinations |
16. Final Revision Checklist
You have mastered this chapter if you can:
- Differentiate permutation vs combination instantly
- Handle identical objects confidently
- Solve circular arrangement problems
- Apply restrictions correctly
- Simplify factorial expressions