This document defines key concepts related to permutations and combinations, including: - Factorials and how they are used to calculate permutations and combinations - The difference between permutations, which consider order, and combinations, which do not - Properties and formulas for calculating permutations and combinations in various scenarios like circular arrangements, some objects being alike, or applying conditions - How to calculate combinations when objects can be selected or arranged multiple times

1. Chapter 5. Basic Concepts of Permutations and Combinations The factorial n, written as n!, represents the product of all integers from 1 to n both inclusive. Hence, n! = n (n – 1) (n – 2) (n – 3)…. 3.2.1 0! = 1 Permutation The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collections of objects with due regard being paid to the order of arrangement or selection, are called permutations. Permutations of n things chosen r at a time, nPr = n (n – 1) (n – 2)… (n – r + 1) Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

2. Chapter 5. Basic Concepts of Permutations and Combinations Properties of Permutation Permutations of n things taken all n things at a time, nPn= n! nPr in factorial notation is written as n! / (n – r)! Circular Permutation The number of circular permutations of n different things chosen at a time is (n – 1)! The number of ways of arranging n persons along a round table so that no person has the same two neighbors = ½ (n – 1)! The number of necklaces formed with n beads of different colors = = ½ (n – 1)! Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

3. Chapter 5. Basic Concepts of Permutations and Combinations Permutations with different conditions applied Number of permutations of n distinct objects when a particular object is not taken in any arrangement is n-1Pr Number of permutations of n distinct objects when a particular object is always included in the arrangement is r. n-1Pr – 1 Combinations The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important are called combinations Number of combinations of n different things taken r at a time = nCr given that 0 ≤ r ≤ n nCr = n! / r! (n – r)! Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

4. Chapter 5. Basic Concepts of Permutations and Combinations Properties of nCr nCr = nCn – r (given that 0 ≤ n – r ≤ n) n+1Cr = nCr + nCr-1 nC0 = 1 nCn = 1 Permutation when some things are alike, all taken at a time P = n! / (p! q! r!) where n is the number of things, p number of the things are exactly of one kind, q of exactly another kind, r of a third kind and the remaining things different. Permutation when each things may be repeated upto r times in an arrangement = nr Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1

5. Chapter 5. Basic Concepts of Permutations and Combinations Combination of n different things taken some or all of n things at a time = 2n – 1 Combination of n things taken some or all at a time when p of the things are alike of one kind, q of the things are alike and of another kind and r of the things are alike of a third kind = [(p + 1) (q + 1)(r + 1)….] – 1 Combination of selecting r1 things from a set of n1 objects and r2 things from a set of n2 objects where combination of r1 things and r2 things are independent is given by n1Cr1 x n2Cr2 Revision Notes – Quantitative Aptitude www.cptsuccess.com Page 1 of 1