NCERT Class 11-Math's: Chapter –7 Permutations and Combinations Part 1

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7.1 Overview

The study of permutations and combinations is concerned with determining the number of different ways of arranging and selecting objects out of a given number of objects, without actually listing them. There are some basic counting techniques which will be useful in determining the number of different ways of arranging or selecting objects. The two basic counting principles are given below:

Fundamental principle of counting

7.1.1 Multiplication principle (Fundamental Principle of Counting)

Suppose an event E can occur in m different ways and associated with each way of occurring of E, another event F can occur in different ways, then the total number of occurrence of the two events in the given order is .

7.1.2 Addition principle

If an event E can occur in m ways and another event F can occur in n ways, and suppose that both can-not occur together, then E or F can occur in ways.

7.1.3 Permutations

A permutation is an arrangement of objects in a definite order.

7.1.4 Permutation of n different objects

The number of permutations of n objects taken all at a time, denoted by the symbol

where n = n(n – 1) (n – 2) ... 3.2.1, read as factorial n, or n factorial.

The number of permutations of n objects taken at a time, where , denoted by , is given by

We assume that

7.1.5 When repetition of objects is allowed The number of permutations of n things taken all at a time, when repletion of objects is allowed is .

The number of permutations of n objects, taken r at a time, when repetition of objects is allowed, is .

7.1.6 Permutations when the objects are not distinct The number of permutations of n objects of which are of one kind, are of second kind, ..., are of kind and the rest if any, are of different kinds is

7.1.7 Combinations On many occasions we are not interested in arranging but only in selecting r objects from given n objects. A combination is a selection of some or all of a number of different objects where the order of selection is immaterial. The number of selections of r objects from the given n objects is denoted by , and is given by

Remarks

1. Use permutations if a problem calls for the number of arrangements of objects and different orders are to be counted.

2. Use combinations if a problem calls for the number of ways of selecting objects and the order of selection is not to be counted.

7.1.8 Some important results

Let n and r be positive integers such that . Then

(i)

(ii)

(iii)

7.2 Solved Examples

Short Answer Type

Question 1:

In a class, there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class for a function. In how many ways can the teacher make this selection?

Answer:

Here the teacher is to perform two operations:

(i) Selecting a boy from among the 27 boys and

(ii) Selecting a girl from among 14 girls.

The first of these can be done in 27 ways and second can be performed in 14 ways. By the fundamental principle of counting, the required number of ways is .

Question 2:

(i) How many numbers are there between and having in the units place?

(ii) How many numbers are there between and having atleast one of their digits ?

Answer:

(i) First note that all these numbers have three digits. 7 is in the unit’s place. The middle digit can be any one of the 10 digits from 0 to 9. The digit in hundred’s place can be any one of the 9 digits from 1 to 9. Therefore, by the fundamental principle of counting, there are numbers between 99 and 1000 having 7 in the unit’s place.

(ii) Total number of 3 digit numbers having atleast one of their digits as 7 = (Total numbers of three digit numbers) – (Total number of 3 digit numbers in which 7 does not appear at all).