Permutation and Combination Concept

Permutation and combination is one of the most important and frequently tested topics in quantitative aptitude for competitive exams. It is asked in almost every major exam including CAT, MBA entrance exams, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. A strong understanding of permutation and combination concept including factorial, nCr, nPr formulas and counting principles is essential for scoring well in these exams. In this post we cover everything from the basic concept of selection and combination using nCr formula, arrangement using nPr formula, circular permutation, formation of geometrical figures, chess board and grid problems, word and number formation, rank of a word in dictionary, division and distribution of objects into groups, number of integral solutions, inclusion exclusion principle and de-arrangement concept — all explained with clear formulas and solved examples.

📚 What You Will Learn in This Post

Selection and Combination — nCr Formula and Solved Examples

Permutation and Arrangement — nPr Formula and Circular Permutation

Word and Number Formation — Rank of Word in Dictionary

Formation of Geometrical Figures — Chess Board and Grid Problems

Division and Distribution of Objects into Groups — All Cases

Number of Integral Solutions and Inclusion Exclusion Principle

De-arrangement Concept — Formula and Solved Examples

1. Permutation & Combination
(i) Introduction of Selection/Combination
(ii) Number of ways of selecting ‘r’ articles from ‘n’ distinct article
(iii) Number of ways of selecting ‘r’ articles from ‘n’ distinct articles if ‘k’ article is always included

2. Total Number of combination of ‘n’ different things taken one or more at a time

3. Selection from identical object/article
(i) The total number of ways of selecting one or more things from ‘p’ identical things of one type, ‘q’ identical things of another type, ‘r’ identical things of another type and ‘n’ distinct things

4. Factor theory concept

5. Exercise

6. Formation of geometrical figures
(i) Number of straight lines from given ‘n’ points
(ii) Number of straight lines from given ‘n’ points, out of them exactly ‘k’ points are collinear
(iii) Number of triangle from given ‘n’ points if no three of them are collinear
(iv) Number of polygon of ‘r’ sides from given ‘n’ points if no three of them are collinear
(v) Number of diagonal of a polygon of n sides

7. Chess board Concept

8. Grid based problem Concept
(i) Number of Rectangle
(ii) Number of Squares
(iii) Shortest path Concept

9. Exercise

1. Permutation or Arrangement
(i) Number of ways in which ‘n’ different objects can be arranged

2. Exercise

3. Circular arrangement
(i) When clockwise ↻ & anticlockwise ↺ arrangement are different then number of arrangement of n different things around a circle.
(ii) When clockwise ↻ & anticlockwise ↺ arrangement are same: i.e. Formation of Garland then number of arrangement of n different things.

4. Exercise

5. Word/Number Formation
(i) Formation of word
(ii) Formation of number

6. Exercise

7. Rank of a word in dictionary

8. Exercise

1. Division & Distribuition of objects into Groups
case(1) → (i) → (A)
Distinct object ━ Distinct Group
┃
(i) Group size Fixed
┃
(A) Group sizes are distinct

case (1) → (i) → (B)
Distinct object ━ Distinct Group
┃
(i) Group size Fixed
┃
(A) Group sizes are equal

case (1) → (i) → (C)
Distinct object ━ Distinct Group
┃
(i) Group size Fixed
┃
(C) Group sizes are mix of distinct & equal

case (1) → (ii)

Distinct object ━ Distinct group
┃
(ii) Group size NOT fixed

case (2): Identical object ━ Distinct group

case (3): Distinct object ━ Identical group

case (4): Identical object ━ Identical groups

2. Exercise

1. Number of Integral Solution

2. Special cases of integral solution

3. Application of multinomial in permutation with repetition

4. Inclusion – Exclusion principle

5. De – arrangement concept

❓ Frequently Asked Questions on Permutation and Combination

Q1. What is the difference between permutation and combination?

Permutation is arrangement of objects where order matters. Combination is selection of objects where order does not matter. nPr = n! divided by (n-r)! gives number of arrangements. nCr = n! divided by r!(n-r)! gives number of selections. Permutation and combination is a fundamental topic in combinatorics asked in CAT, SSC CGL, Bank PO and Railway exams.

Q2. What is nCr and nPr formula?

nCr = n! divided by r! × (n-r)! where n is total objects and r is objects to be selected. nPr = n! divided by (n-r)! where n is total objects and r is objects to be arranged. nPr = nCr × r! because after selecting r objects from n we can arrange them in r! ways.

Q3. What is circular permutation formula?

When n different objects are arranged in a circle and clockwise and anticlockwise arrangements are different then number of arrangements = (n-1)!. When clockwise and anticlockwise are same as in formation of garland then number of arrangements = (n-1)! divided by 2. Circular permutation problems are very commonly asked in CAT and MBA entrance exams.

Q4. How to find rank of a word in dictionary?

To find rank of a word in dictionary arrange all letters alphabetically. For each position count how many letters are smaller than the current letter and multiply by factorial of remaining positions. Add all such values and add 1 to get the rank. This method gives the position of the word if all arrangements of its letters were listed alphabetically.

Q5. What is de-arrangement concept?

De-arrangement is the number of permutations of n objects in which no object appears in its original position. Formula = n! × (1 – 1/1! + 1/2! – 1/3! + … + (-1)^n/n!). For example de-arrangement of 3 objects = 2, of 4 objects = 9. This concept is very important for CAT and advanced competitive exams.

Q6. What is division and distribution of objects into groups?

Division and distribution covers four main cases — distinct objects into distinct groups, identical objects into distinct groups, distinct objects into identical groups and identical objects into identical groups. Each case has different formulas based on whether group sizes are fixed or not. This is one of the most important and complex topics in permutation and combination for CAT.

Q7. Which competitive exams cover permutation and combination?

Permutation and combination is asked heavily in CAT, XAT, MBA entrance exams, SSC CGL, Bank PO, Railway RRB and CSAT. It is closely related to Probability and Counting so mastering all three together is highly recommended.

Q8. Where can I practice permutation and combination questions?

After understanding the concept you can practice on our Permutation and Combination Exercise page which contains solved practice questions covering all types of problems asked in competitive exams. You can also check our Probability Concept and Counting Concept pages for related topics.

📚 What You Will Learn in This Post

❓ Frequently Asked Questions on Permutation and Combination

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