🔹 1. Introduction

This chapter deals with counting techniques—how to arrange and select objects efficiently.

👉 Key idea:

• Permutation → Arrangement (order matters)
• Combination → Selection (order does NOT matter)

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🔹 2. Fundamental Principle of Counting

✔️ Multiplication Principle

If a task can be done in:

• m ways AND another in n ways

👉 Total ways = m × n

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✔️ Addition Principle

If a task can be done in:

• m ways OR n ways

👉 Total ways = m + n

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🔹 3. Factorial Notation

✔️ Definition

n! = n × (n−1) × … × 1

Special cases:

• 0! = 1
• 1! = 1

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🔹 4. Permutations

✔️ Definition

Arrangement of objects in a specific order.

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✔️ Formula

Number of permutations of n objects taken r at a time:

$^{n}P_r = \frac{n!}{(n-r)!}$nPr​=(n−r)!n!​

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✔️ Special Case

Arrangement of all objects:

$^{n}P_n = n!$nPn​=n!

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🔹 5. Permutations with Repetition

If repetition is allowed:

$n^r$nr

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✔️ Example

Number of 3-digit numbers using digits 1–5:

= 5³ = 125

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🔹 6. Circular Permutations

✔️ Formula

$(n-1)!$(n−1)!

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✔️ Notes

• Used when arrangement is around a circle
• Clockwise and anticlockwise considered same

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🔹 7. Permutations of Identical Objects

If n objects where:

• p are identical
• q are identical

Then:

$\frac{n!}{p!q!}$p!q!n!​

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✔️ Example

Word: “BALL”

= 4! / 2! = 12

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🔹 8. Combinations

✔️ Definition

Selection of objects where order does not matter.

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✔️ Formula

$^{n}C_r = \frac{n!}{r!(n-r)!}$nCr​=r!(n−r)!n!​

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✔️ Relation with Permutation

$^{n}P_r = ^{n}C_r \cdot r!$nPr​=nCr​⋅r!

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🔹 9. Important Properties of Combinations

✔️ Symmetry

$^{n}C_r = ^{n}C_{n-r}$nCr​=nCn−r​

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✔️ Pascal Identity

$^{n}C_r + ^{n}C_{r-1} = ^{n+1}C_r$nCr​+nCr−1​=n+1Cr​

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✔️ Sum of All Combinations

\sum_{r=0}^{n} ^{n}C_r = 2^n

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🔹 10. Important Types of Problems

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✔️ 1. Selection Problems

Example:
Select 3 students from 10:

= ¹⁰C₃

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✔️ 2. Arrangement with Restrictions

Example:
People must sit together → treat as one unit

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✔️ 3. Distribution Problems

Example:
Distribute objects into boxes

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✔️ 4. Digit Formation

• With repetition → n^r
• Without repetition → nPr

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🔹 11. Key Differences

Permutation	Combination
Order matters	Order doesn’t matter
nPr	nCr
Arrangement	Selection

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🔹 12. Important Identities

✔️ 1

$^{n}C_0 = ^{n}C_n = 1$nC0​=nCn​=1

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✔️ 2

$^{n}C_1 = n$nC1​=n

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✔️ 3

$^{n}C_r = ^{n-1}C_r + ^{n-1}C_{r-1}$nCr​=n−1Cr​+n−1Cr−1​

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🔹 13. Applications

✔️ Probability
✔️ Number systems
✔️ Arrangements
✔️ Coding-decoding
✔️ JEE problem solving

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🔹 14. JEE & CBSE Important Points

✔️ Understand difference between nPr and nCr
✔️ Practice restriction-based problems
✔️ Circular permutation is important
✔️ Repetition vs no repetition is key
✔️ Identity-based questions common

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🔹 15. Common Mistakes

❌ Using permutation instead of combination
❌ Ignoring identical objects
❌ Wrong factorial simplification
❌ Not considering restrictions

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🔹 16. Practice Questions

1. Find number of permutations of 5 objects
2. Select 4 students from 12
3. Arrange letters of “BANANA”
4. Circular seating problems
5. Form numbers using digits

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🔹 17. Quick Revision Sheet

• nPr = n! / (n−r)!
• nCr = n! / r!(n−r)!
• nPr = nCr × r!
• Circular = (n−1)!
• Repetition = n^r

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🔹 18. Conclusion

Permutations and Combinations is a high-weightage chapter in JEE and CBSE. With proper understanding and practice, it becomes one of the most scoring topics.