What is the difference between permutation and combination?

Permutation is about arrangement — order matters. ABC ≠ BAC. Combination is about selection — order doesn't matter. {A,B,C} = {B,A,C}. Use permutation for queues, passwords; use combination for committees, selections.

By definition: n! = n × (n−1)! So 1! = 1 × 0! → 0! = 1!/1 = 1. Also, the number of ways to arrange 0 objects is exactly 1 (doing nothing).

How do you solve 'at least' combination problems?

Use complementary counting: P(at least 1) = Total possibilities − P(none selected). For example, at least one woman in a group = (all possible groups) − (groups with no women).

Fundamental Counting Principles

Permutations and Combinations • Class 11 Mathematics • NCERT • CBSE

Permutation nPr = n!/(n-r)!. Combination nCr = n!/r!(n-r)!. Multiplication rule: if task A has m ways and task B has n ways then together they have m×n ways. nCr = nC(n-r). nC0 = nCn = 1.

Key Formulas

ⁿPᵣ = n! / (n−r)!
ⁿCᵣ = n! / [r!(n−r)!]
ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎
ⁿPᵣ = ⁿCᵣ × r!
Circular permutations = (n−1)!
ⁿCᵣ + ⁿCᵣ₋₁ = ⁽ⁿ⁺¹⁾Cᵣ (Pascal's identity)

Frequently Asked Questions

What is the difference between permutation and combination?: Permutation is about arrangement — order matters. ABC ≠ BAC. Combination is about selection — order doesn't matter. {A,B,C} = {B,A,C}. Use permutation for queues, passwords; use combination for committees, selections.
Why is 0! = 1?: By definition: n! = n × (n−1)! So 1! = 1 × 0! → 0! = 1!/1 = 1. Also, the number of ways to arrange 0 objects is exactly 1 (doing nothing).
How do you solve 'at least' combination problems?: Use complementary counting: P(at least 1) = Total possibilities − P(none selected). For example, at least one woman in a group = (all possible groups) − (groups with no women).

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