Schooling · Class 11 · Mathematics · Permutations and Combinations · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Permutations and Combinations: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Fundamental Principle of Counting
- Factorial notation
- Permutation
- Combination
- Relation between permutations and…
- Permutations with repetition
- Combinations have symmetry
- Formulas and facts
- Factorial
- Permutation formula
- Combination formula
- Permutations of n objects where p…
- Sum property
- Pascal's identity
- Special values
- Worked examples
- Forming three-digit numbers
- Selecting a committee
- Arranging letters with repetition
- Watch out for
- Confusing permutations with…
- Forgetting that 0! = 1
- Using P(n, r) when objects repeat
- Miscounting in at least or at most…
- Applying C(n, r) with r greater than n
What this chapter is about
This chapter introduces systematic methods for counting the number of ways to arrange or select objects from a given set. When you need to find how many different seating arrangements are possible for a group of people, or how many ways you can choose a committee from a class, you are dealing with counting problems that become tedious if done by listing every possibility. Permutations and combinations provide formulas that handle such problems efficiently.
Key ideas
- Fundamental Principle of Counting: If one event can occur in m ways and, after it occurs, a second event can occur in n ways, then the two events together can occur in m × n ways. This extends to any number of successive events.
- Factorial notation: n! (read as n factorial) equals n × (n−1) × (n−2) × ... × 3 × 2 × 1 for any positive integer n. By definition, 0! = 1.
- Permutation: A permutation is an arrangement of objects in a definite order. The number of permutations of n distinct objects taken r at a time is denoted P(n, r) or ⁿPᵣ.
- Combination: A combination is a selection of objects where order does not matter. The number of combinations of n distinct objects taken r at a time is denoted C(n, r) or ⁿCᵣ or (n choose r).
- Relation between permutations and combinations: P(n, r) = C(n, r) × r!, because each combination of r objects can be arranged in r! ways to give permutations.
- Permutations with repetition: When certain objects are identical, divide by the factorials of their repetition counts to avoid overcounting.
- Combinations have symmetry: C(n, r) = C(n, n−r), which means choosing r objects is equivalent to leaving out n−r objects.
Formulas and facts to remember
- Factorial: n! = n × (n−1) × (n−2) × ... × 1; also 0! = 1 and 1! = 1.
- Permutation formula: P(n, r) = n! / (n−r)!, valid for 0 ≤ r ≤ n.
- Combination formula: C(n, r) = n! / [r! × (n−r)!], valid for 0 ≤ r ≤ n.
- Permutations of n objects where p are alike, q are alike, etc.: Number of arrangements = n! / (p! × q! × ...).
- Sum property: C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n) = 2ⁿ.
- Pascal's identity: C(n, r) + C(n, r−1) = C(n+1, r).
- Special values: P(n, n) = n!, P(n, 1) = n, C(n, 0) = 1, C(n, n) = 1.
Common mistakes
- Confusing permutations with combinations → Ask yourself: does the order of selection matter? If yes, use permutation; if no, use combination.
- Forgetting that 0! = 1 → This is a definition, not a calculation; accept it and use it in formulas.
- Using P(n, r) when objects repeat → When identical objects exist, use n! / (p! × q! × ...) instead of the standard permutation formula.
- Miscounting in at least or at most problems → Break into mutually exclusive cases, calculate each, then add.
- Applying C(n, r) with r greater than n → This is undefined; always verify that r ≤ n before using the formula.
Quick revision
- Permutation = arrangement (order matters); Combination = selection (order does not matter).
- P(n, r) = n! / (n−r)!; C(n, r) = n! / [r! × (n−r)!].
- For repeated objects in a word or set, divide n! by the factorials of the repetition counts.
- C(n, r) = C(n, n−r) — choosing r is the same as leaving n−r.
- Always identify whether the problem involves arrangement or selection before choosing a formula.