120. 5 × 4 × 3 × 2 × 1 = 120.

The value of ¹⁰C₈ is:

45. ¹⁰C₈ = ¹⁰C₂ = 45.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 6: Permutations and Combinations

Short notes, key concepts, quick revision and practice MCQs with answers

Counting techniques let us count arrangements and selections without listing them. The chapter covers the fundamental principles of counting, the factorial function, permutations (including repeated and identical objects and circular arrangements) and combinations with their properties.

Key concepts: Permutations and Combinations

Fundamental principles: Addition principle: if one event can occur in m ways and another (with no common outcome) in n ways, either can occur in m + n ways. Multiplication principle: m ways followed by n ways gives m × n ways.
Factorial: n! = 1 × 2 × 3 × ... × n; 0! = 1; n! = n × (n − 1)!.
Permutations: An ordered arrangement. The number of permutations of r objects from n distinct objects is ⁿPᵣ = n!/(n − r)!; ⁿPₙ = n!.
Repetition allowed: When objects may be repeated, r places can be filled from n objects in nʳ ways.
Identical objects: Arrangements of n objects of which p are alike of one kind, q of another and r of a third kind = n!/(p! q! r!).
Circular permutations: n distinct objects around a circle: (n − 1)!. If clockwise and anticlockwise arrangements are the same (garlands, necklaces): (n − 1)!/2.
Combinations: A selection without regard to order. ⁿCᵣ = n!/(r!(n − r)!) and ⁿPᵣ = r! × ⁿCᵣ.
Properties of combinations: ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ; if ⁿCₓ = ⁿCᵧ then x = y or x + y = n; ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ.

Quick revision: important points

Order matters: permutation. Order does not matter: combination.
ⁿC₀ = ⁿCₙ = 1 and ⁿC₁ = n.
Number of handshakes among n people = ⁿC₂.
Number of diagonals of an n-sided polygon = ⁿC₂ − n.
Number of lines through n points (no three collinear) = ⁿC₂; triangles = ⁿC₃.

Practice MCQs with answers

The value of 5! is:
1. 20
2. 60
3. 120
4. 720
Show answer
Answer: C. 120
5 × 4 × 3 × 2 × 1 = 120.
The value of 0! is:
1. 0
2. 1
3. Not defined
4. ∞
Show answer
Answer: B. 1
By definition 0! = 1.
The value of ⁵P₂ is:
1. 10
2. 20
3. 25
4. 60
Show answer
Answer: B. 20
5!/3! = 5 × 4 = 20.
The value of ⁶C₂ is:
1. 12
2. 15
3. 30
4. 36
Show answer
Answer: B. 15
6 × 5/2 = 15.
The value of ¹⁰C₈ is:
1. 45
2. 90
3. 80
4. 10
Show answer
Answer: A. 45
¹⁰C₈ = ¹⁰C₂ = 45.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters