Chapter 6 — Permutations and Combinations

Chapter 6 of the Class 11 Maths NCERT textbook, "Permutations and Combinations", covers counting techniques, factorial notation, and formulas for arranging and selecting objects without listing them explicitly.

• Counting without listing — The chapter builds on the Fundamental Principle of Counting to solve large counting problems systematically, so you can find how many outcomes exist without ever writing them all out — the core payoff of this topic.
• Order matters vs order doesn't — It draws the central distinction between permutations, which count ordered arrangements, and combinations, which count unordered selections, and links them through the relationship that a permutation is a combination times the arrangements of the chosen items.
• Refinements: repetition and identical objects — Beyond the basic formulas, the chapter handles cases where repetition is allowed or where some objects are identical, adjusting the count so the same techniques cover words, seating and committee problems accurately.

1. 01Fundamental Principle of Counting: If an event can occur in m ways and another in n ways, the total combined occurrences = m × n
2. 02Factorial notation: n! = n × (n-1) × (n-2) × ... × 1, with 0! = 1 by definition
3. 03Permutations (nPr) count ordered arrangements; formula: nPr = n!/(n-r)! for 0 ≤ r ≤ n
4. 04Combinations (nCr) count unordered selections; formula: nCr = n!/[r!(n-r)!]; satisfies nCn-r = nCr
5. 05Permutations with repetition allowed: nr (each of r positions can be filled r ways)
6. 06When objects are not all distinct, divide by factorials of repeated objects: e.g., permutations of ROOT = 4!/2! = 12