Summary
Permutations and Combinations is a Class 11 NCERT Mathematics chapter covering counting techniques, factorial notation, and formulas for arranging and selecting objects without listing them explicitly.
Chapter 6 introduces the Fundamental Principle of Counting (multiplication principle), which enables solving complex counting problems systematically. It covers factorial notation (n! = 1 × 2 × 3 × ... × n), permutations (ordered arrangements) with the formula nPr = n!/(n-r)!, and combinations (unordered selections) with nCr = n!/[r!(n-r)!]. The chapter includes theorems for permutations with repetition allowed (nr) and when objects are not all distinct, plus the relationship between permutations and combinations: nPr = nCr × r!. Applications include forming numbers, seating arrangements, committee selection, and card combinations.
Key points & formulas
- 01Fundamental Principle of Counting: If an event can occur in m ways and another in n ways, the total combined occurrences = m × n
- 02Factorial notation: n! = n × (n-1) × (n-2) × ... × 1, with 0! = 1 by definition
- 03Permutations (nPr) count ordered arrangements; formula: nPr = n!/(n-r)! for 0 ≤ r ≤ n
- 04Combinations (nCr) count unordered selections; formula: nCr = n!/[r!(n-r)!]; satisfies nCn-r = nCr
- 05Permutations with repetition allowed: nr (each of r positions can be filled r ways)
- 06When objects are not all distinct, divide by factorials of repeated objects: e.g., permutations of ROOT = 4!/2! = 12
Frequently asked questions
01What is the difference between permutations and combinations?
Permutations count ordered arrangements where sequence matters (e.g., ABC ≠ BAC). Combinations count unordered selections where sequence does not matter (e.g., selecting 2 students from 5). Permutations use the formula nPr = n!/(n-r)!, while combinations use nCr = n!/[r!(n-r)!]. Mathematically, nPr = nCr × r!.
02What is the Fundamental Principle of Counting and how is it used?
It states: if event A can occur in m ways and event B can occur in n ways, then both events together occur in m × n ways. For example, if you have 3 pants and 2 shirts, you can form 3 × 2 = 6 different outfits. It generalizes to any finite number of successive events: for 3 events occurring in m, n, and p ways respectively, total ways = m × n × p. This principle underlies all permutation and combination problems.
03Is the NCERT Class 11 Maths Chapter 6 PDF free to download?
Yes, the NCERT Class 11 Mathematics Chapter 6 PDF is free to download. It is publicly available as part of NCERT's free educational resources for Class 11 students preparing for CBSE exams.
04How do you calculate permutations when repetition is allowed?
When repetition is allowed, the formula is nr, where n is the total number of objects and r is the number of positions to fill. For example, forming 4-letter words from 4 letters with repetition allowed = 4^4 = 256. This is because each of the 4 positions can be filled with any of the 4 letters independently. Without repetition, the count would be 4! = 24.
More chapters in Mathematics
This is the complete Mathematics Chapter 6 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all NCERT Class 11 textbooks.
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