Permutations and Combinations Class 11: Notes & Questions

Revision notes

Permutations and Combinations: revision notes

1. Fundamental principle of counting

If one event can happen in $m$ ways and, after it, another in $n$ ways, the two together can happen in $m n$ ways (multiplication principle). Use "and" $\to$ multiply; "or" (cases that cannot both happen) $\to$ add.

2. Factorial

$n! = 1 \times 2 \times \dots \times n$ , $0! = 1$ , $n! = n \times (n - 1)!$ . Simplify quotients before multiplying: $\frac{8!}{6!} = 8 \times 7 = 56$ .

3. Permutations (order matters)

Arrangements of $r$ of $n$ different objects: $^{n} P_{r} = \frac{n!}{(n - r)!}$ ; all $n$ : $n!$ .
With repetition allowed: $n^{r}$ .
$n$ objects with $p$ alike of one kind, $q$ alike of another, ...: $\frac{n!}{p! q! \dots}$ .
Objects that must stay together: treat them as one block, then arrange inside the block.

4. Combinations (order does not matter)

Selections of $r$ from $n$ different objects: $^{n} C_{r} = \frac{n!}{r! (n - r)!} = \frac{^{n} P_{r}}{r!}$ . Properties: $^{n} C_{r} =^{n} C_{n - r}$ ; $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ ; if $^{n} C_{a} =^{n} C_{b}$ then $a = b$ or $a + b = n$ .

5. Choosing the method

Arranging, ordering, forming numbers or words $\to$ permutations. Choosing a team, committee or group $\to$ combinations. Mixed problems: first select (combinations), then arrange (permutations). Deal with restrictions (a digit that cannot be first, people who must be included) first.

Worked example 1

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

$^{5} P_{3} = 5 \times 4 \times 3 = 60$ .

Worked example 2

In how many ways can a committee of $3$ be chosen from $9$ people?

$^{9} C_{3} = \frac{9 \times 8 \times 7}{6} = 84$ .

Worked example 3

How many arrangements are there of the letters of LEVEL?

$\frac{5!}{2! 2!} = 30$ (two L's, two E's).

Common errors

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").

Exam tips

Draw boxes for the positions and fill in the number of choices for each, restricted positions first.
Write the formula with numbers, e.g. $^{10} C_{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1}$ , then simplify.

Topics in this chapter: Factorial · Permutations · Combinations · Fundamental principle of counting.

Formulas for this chapter: Chapter 6 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can use the multiplication principle, simplify factorials and apply nPr and nCr directly.

Read first: 1. Fundamental principle; 2. Factorial; 3. Permutations; 4. Combinations 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can solve for n in nPr / nCr equations, count words with restrictions and form committees.

Read first: 3. Alike objects and blocks; 4. Properties of nCr; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can combine selection and arrangement in multi-part problems about words, numbers and teams.

Read first: 5. Choosing the method 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle divisibility and digit restrictions and 'must include / must exclude' selections.

Read first: 5. Restrictions first; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 32 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceFactorial

The value of $\frac{8!}{6!}$ is

(a) $2$
(b) $48$
(c) $56$
(d) $\frac{4}{3}$

Q2·1 mark·Multiple choicePermutations

The number of 3-digit numbers with distinct digits that can be formed from $1, 2, 3, 4, 5$ is

(a) $10$
(b) $60$
(c) $125$
(d) $15$

Q3·1 mark·Multiple choiceCombinations

If $^{n} C_{2} = 45$ , then $n$ is

(a) $10$
(b) $9$
(c) $15$
(d) $45$

Worked solution·2 marks·Very short answerPermutations

Find $n$ if $^{n} P_{4} = 20 \times^{n} P_{2}$ .

Full worked solution Free, no sign-in

$n (n - 1) (n - 2) (n - 3)$ $= 20 n (n - 1) \Rightarrow (n - 2) (n - 3)$ $= 20$ M1
$n^{2} - 5 n - 14 = 0 \Rightarrow n = 7$ ( $n \geq 4$ ) A1

Every question here has a full worked solution. See them all →

Stretch yourself: original combinatorics problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S13 · Problem S61 · Problem S12 · Problem S60. All 22 →

Knowledge organiser: Permutations and Combinations on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: nCr formula. All formulas A–Z

Where marks are lost in Permutations and Combinations

Permutation used for a selection (or combination for an arrangement). Fix: ask 'does the order matter?' before choosing nPr or nCr.
Zero allowed in the first place of a number. Fix: fill the first box first, with the non-zero digits.
Repeated letters not divided out. Fix: n!/(p! q! …) for alike objects.
Block method without the internal arrangements. Fix: multiply by the ways to arrange the block itself.
Cases multiplied instead of added. Fix: separate cases (last digit 0, or last digit 2) are added.

Test Permutations and Combinations against the clock

Want it against the clock? Take a timed 30-mark chapter test on Permutations and Combinations, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE? Try the JEE-style questions on permutations, combinations and the binomial theorem.

Permutations and Combinations: common questions

What is in CBSE Class 11 Maths Chapter 6, Permutations and Combinations?

The topics are: Factorial, Permutations, Combinations, Fundamental principle of counting. This page has revision notes for each, and 32 board-style practice questions with step-marked solutions.

How many marks is Permutations and Combinations worth?

It is in the Algebra unit, which carries 25 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Permutations and Combinations?

Permutation used for a selection (or combination for an arrangement). Fix: ask 'does the order matter?' before choosing nPr or nCr. Zero allowed in the first place of a number. Fix: fill the first box first, with the non-zero digits.

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Permutations and Combinations Class 11: notes and important questions

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