Permutations and Combinations Class 11 Math Quick Recall / Short Notes (2026-27)

The Art of Arrangement and Selection: Permutations and Combinations

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Fundamental Principle of Counting (FPC):

And (Multiplication): If task A in m ways AND task B in n ways -> m × n ways.

Or (Addition): If task A in m ways OR task B in n ways -> m + n ways.

Factorial (n!): n! = n × (n-1) × ... × 1. (0! = 1).

Permutation (nPr): Arrangement where order MATTERS. nPr = n! / (n - r)!.

Combination (nCr): Selection where order DOES NOT matter. nCr = n! / [r!(n - r)!].

Relation: nPr = nCr × r!. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Permutations and Combinations provide the mathematical toolkit for counting possibilities and arrangements in complex systems. Master the Fundamental Principle of Counting, nPr for ordered arrangements, and nCr for selections to excel in probability and cryptography. This Class 11 Math Chapter 7 guide covers all essential factorial logic for JEE and CBSE exams. Mathematics is not just about numbers; it's about possibilities.

1. Fundamental Principle of Counting (FPC)

This is the base of all counting techniques.

Multiplication Principle: If an event occurs in m different ways, following which another event occurs in n different ways, then the total number of occurrence of the events in the given order is m × n.
Addition Principle: If an event can occur in m ways and another independent event can occur in n ways, then either of the two events can occur in m + n ways.

2. Factorials (n!)

The product of first n natural numbers is called n-factorial.

n! = 1 × 2 × 3 × ... × n.
0! = 1 (by definition).
n! = n × (n - 1)!.

3. Permutations (Arrangements)

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

Theorem 1: The number of permutations of n different objects taken r at a time (0 < r ≤ n) and objects do not repeat is nPr = n! / (n - r)!.
Theorem 2: If repetition is allowed, the number of permutations is nʳ.
Theorem 3: If out of n objects, p are of one kind, q of another, and the rest are different, number of permutations = n! / (p!q!).

4. Combinations (Selections)

A combination is a selection of items where the order of selection does not matter.

Theorem: The number of combinations of n different objects taken r at a time is nCr = n! / [r!(n - r)!].
Properties:
1. nCr = nC(n-r)
2. nCa = nCb => either a = b or a + b = n.
3. nCr + nC(r-1) = (n+1)Cr (Pascal's Formula).

5. Difference: Permutation vs Combination

Feature	Permutation	Combination
Focus	Arrangement / Order	Selection / Grouping
Order	Matters	Does not matter
Keyword	Arrange, List, Align	Select, Choose, Pick
Formula	nPr	nCr

Comprehensive Exam Strategy (Q&A)

Q1: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5 assuming that repetition of digits is allowed? Answer:

Total digits = 5. Places to fill = 3.
Using FPC: 5 × 5 × 5 = 125 ways.

Q2: Find n if n-1P3 : nP4 = 1 : 9. Answer:

[(n-1)! / (n-1-3)!] / [n! / (n-4)!] = 1/9
[(n-1)! / (n-4)!] × [(n-4)! / n!] = 1/9
(n-1)! / n! = 1/9
(n-1)! / n(n-1)! = 1/9
n = 9.

Q3: A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? Answer:

Order doesn't matter, so use combinations.
Total people = 2 + 3 = 5. Select 3.
5C3 = 5! / (3!2!) = (5 × 4) / 2 = 10 ways.

Related Revision Notes

Chapter 6: Linear Inequalities
Chapter 8: Binomial Theorem
[External Reference: NCERT Class 11 Math Chapter 7 (Authoritative Source)]

Conclusion

Permutations and Combinations transform the way we see complexity. By mastering the core formulas of nPr and nCr, and understanding when order matters, you gain the power to calculate outcomes in everything from poker hands to the number of ways to sequence DNA. Keep your factorials small and your logic sharp!