Knowledge organisers

Permutations and Combinations: CBSE Class 11 Maths knowledge organiser

Everything to know about permutations and combinations on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Fundamental principle of counting: If one thing can happen in m ways and then another in n ways, both can happen in mn ways.
Permutation: An arrangement where order matters: ⁿPᵣ = n!/(n − r)!.
Combination: A selection where order does not matter: ⁿCᵣ = n!/(r!(n − r)!).

Key formulas

Arrangements; selections	\({}^nP_r=\frac{n!}{(n-r)!},\) \({}^nC_r=\frac{n!}{r!(n-r)!}\)
Alike objects	\(\frac{n!}{p!\,q!\cdots}\)
Properties	\({}^nC_r={}^nC_{n-r},\) \({}^nC_r+{}^nC_{r-1}={}^{n+1}C_r\)

Worked example

How many 3-digit numbers with distinct digits can be formed from \(1, 2, 3, 4, 5\)?

\({}^5P_3 = 5 \times 4 \times 3 = 60\).

Common mistakes

Using combinations when order matters (or the reverse).
Letting \(0\) be the first digit of a number.
Forgetting to divide by \(p!\) for repeated letters, or to multiply by the arrangements inside a block.
Adding when the principle needs multiplying ("and"), or multiplying separate cases ("or").

You should be able to…

Use the multiplication principle, simplify factorials and apply nPr and nCr directly.
Solve for n in nPr / nCr equations, count words with restrictions and form committees.
Combine selection and arrangement in multi-part problems about words, numbers and teams.
Handle divisibility and digit restrictions and 'must include / must exclude' selections.

The printable sheet

Permutations and Combinations knowledge organiser for CBSE Class 11 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Permutations and Combinations knowledge organiser (CBSE Class 11 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 11 Maths topics: Sets · Relations and Functions · Trigonometric Functions · Complex Numbers and Quadratic Equations · Linear Inequalities · Binomial Theorem · Sequences and Series · Straight Lines · Conic Sections · Introduction to Three Dimensional Geometry · Limits and Derivatives · Statistics · Probability · All CBSE Class 11 Maths organisers