NCERT Mathematics Class 11 - Chapter 6: Permutations and Combinations - Notes

Learning Objectives

• Understand the fundamental counting principles
• Learn permutations (arrangements) and their formulas
• Learn combinations (selections) and their formulas
• Apply PnC to solve counting problems

Key Concepts

Fundamental Principles of Counting

Multiplication Principle: If event A can occur in m ways and event B can occur in n ways, then A and B together can occur in m × n ways. Addition Principle: If event A can occur in m ways OR event B can occur in n ways (mutually exclusive), then either can occur in m + n ways.

Factorial

n! = n × (n-1) × (n-2) × ... × 2 × 1. 0! = 1 (by convention). 1! = 1. n! = n × (n-1)!. Factorial is defined only for non-negative integers.

Permutations (Arrangements)

Number of arrangements of n objects taken r at a time: P(n,r) = n!/(n-r)!. P(n,n) = n! (all objects arranged). P(n,1) = n.

Permutations with repetition: n^r (r positions, each with n choices). Permutations of n objects with identical objects: n!/(p! × q! × ...) where p, q, ... are frequencies of identical objects. Example: Arrangements of letters in "MISSISSIPPI" = 11!/(4!×4!×2!).

Circular permutations: (n-1)! for n distinct objects in a circle. If clockwise and anticlockwise are same (necklace/bracelet): (n-1)!/2.

Combinations (Selections)

Number of ways to select r objects from n: C(n,r) = n!/(r!(n-r)!). C(n,0) = C(n,n) = 1. C(n,r) = C(n,n-r) (complementary property). C(n,1) = n.

Key identity: C(n,r) + C(n,r-1) = C(n+1,r) (Pascal's rule).

Important results: C(n,0) + C(n,1) + ... + C(n,n) = 2ⁿ. Number of ways to distribute n identical items into r distinct groups (at least 0 each) = C(n+r-1, r-1) (stars and bars).

Key Differences

Permutation: Order matters (arrangements). Combination: Order doesn't matter (selections). P(n,r) = C(n,r) × r! (each combination can be arranged in r! ways).

Summary

Counting problems use multiplication and addition principles. Permutations count arrangements where order matters. Combinations count selections where order does not matter. Special cases include circular permutations and permutations with repeated objects.

Important Terms

• Factorial (n!): Product of all positive integers up to n; 0! = 1
• Permutation P(n,r): Number of arrangements of r objects from n = n!/(n-r)!
• Combination C(n,r): Number of selections of r objects from n = n!/(r!(n-r)!)
• Circular permutation: Arrangements in a circle = (n-1)!
• Pascal's rule: C(n,r) + C(n,r-1) = C(n+1,r)

Quick Revision

• P(n,r) = n!/(n-r)!; C(n,r) = n!/(r!(n-r)!)
• P(n,r) = C(n,r) × r!
• With repetition: arrangements = n^r
• Identical objects: n!/(p!q!...)
• Circular: (n-1)!; with necklace: (n-1)!/2
• C(n,r) = C(n, n-r); Sum of all C(n,r) = 2ⁿ
• 0! = 1; C(n,0) = 1