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Permutations and Combinations – Class 11 Mathematics notes, important questions and MCQs

NCERT Class 11 Mathematics, chapter 6. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Counting principles find the number of ways to arrange (permutations) or select (combinations) objects without listing them.

Counting and permutations

Fundamental principle: if one task can be done in m ways and another in n ways, both together in m × n ways.
n! = n(n − 1)…1; 0! = 1.
ⁿPᵣ = n!/(n − r)!; arrangements with repeated objects = n!/(p! q! …).

Combinations

ⁿCᵣ = n!/[r!(n − r)!]; ⁿCᵣ = ⁿCₙ₋ᵣ.
ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.
Order matters → permutation; order does not matter → combination.

Key terms

Permutation: an ordered arrangement
Combination: an unordered selection
Factorial notation: n! = 1 × 2 × 3 × … × n, the number of ways to arrange n distinct objects

Formulas

ⁿPᵣ = n!/(n − r)!
ⁿCᵣ = n!/[r!(n − r)!]

Important questions with answers

Find n if ⁿP₂ = 90.2 marks

n(n − 1) = 90, so n = 10.

In how many ways can the letters of the word "INDIA" be arranged? How many begin with I?3 marks

5 letters with I twice: 5!/2! = 60. Beginning with I: arrange N, D, I, A freely: 4! = 24.

How many words, with or without meaning, can be formed from the letters of "EQUATION" using each letter once? How many of them have all vowels together, and how many begin with a vowel?5 marks

EQUATION has 8 distinct letters: 8! = 40,320 words. Vowels E, U, A, I, O (5) together as one block with Q, T, N: 4 units arranged in 4! ways and vowels inside in 5! ways: 24 × 120 = 2,880. Beginning with a vowel: 5 choices for the first letter × 7! for the rest = 5 × 5,040 = 25,200.

Quick revision

0! = 1.
ⁿCᵣ = ⁿCₙ₋ᵣ.
Arrangement = P; selection = C.

Exam tips

Number of diagonals of an n-gon = ⁿC₂ − n.
Decide first whether order matters; "arrange, form numbers, seat" → permutation; "choose, select, form a team" → combination.

Practice MCQs

5! equals
1. 20
2. 60
3. 25
4. 120
Show answer
Answer: (d) 120. 5 × 4 × 3 × 2 × 1.
⁵C₂ equals
1. 10
2. 20
3. 5
4. 25
Show answer
Answer: (a) 10. 5 × 4/2.
Number of arrangements of the letters of the word "BOOK" is
1. 12
2. 24
3. 6
4. 4
Show answer
Answer: (a) 12. 4!/2!.
In how many ways can 3 students be chosen from 7
1. 21
2. 210
3. 35
4. 42
Show answer
Answer: (c) 35. ⁷C₃.
How many 3-digit numbers can be formed from 1, 2, 3, 4, 5 without repetition
1. 120
2. 125
3. 10
4. 60
Show answer
Answer: (d) 60. ⁵P₃.
Number of diagonals of a hexagon is
1. 6
2. 15
3. 9
4. 12
Show answer
Answer: (c) 9. 15 − 6.
If ⁿC₈ = ⁿC₆, then n is
1. 2
2. 14
3. 48
4. 7
Show answer
Answer: (b) 14. 8 + 6.
In how many ways can a committee of 3 men and 2 women be chosen from 6 men and 4 women
1. 120
2. 60
3. 240
4. 210
Show answer
Answer: (a) 120. ⁶C₃ × ⁴C₂ = 20 × 6.

All chapters of Class 11 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.