NCERT Class 11-Math's: Chapter –7 Permutations and Combinations Part 3

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Question 9:

Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.

Answer:

Let us denote married couples by , where each couple is considered to be a single unit as shown in the following figure:

Question 9-Image of Single Unit

Image of Single Unit

Then the number of ways in which spouses can be seated next to each other is ways.

Again each couple can be seated in ways. Thus the total number of seating arrangement so that spouses sit next to each other .

Again, if three ladies sit together, then necessarily three men must sit together. Thus, ladies and men can be arranged altogether among themselves in ways. Therefore, the total number of ways where ladies sit together is .

Question 10:

In a small village, there are 87 families, of which 52 families have almost 2 children. In a rural development programme 20 families are to be chosen for assistance, of which atleast 18 families must have at most 2 children. In how many ways can the choice be made?

Answer:

It is given that out of 87 families, 52 families have at most 2 children so other 35 families are of other type. According to the question, for rural development programme, 20 families are to be chosen for assistance, of which at least 18 families must have almost 2 children. Thus, the following are the number of possible choices:

(18 families having at most 2 children and 2 selected from other type of families)

(19 families having at most 2 children and 1 selected from other type of families)

(All selected 20 families having at most 2 children)

Hence, the total number of possible choices is

Question 11:

A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics Part II, unless Mathematics Part I is also borrowed. In how many ways can he choose the three books to be borrowed?

Answer:

Let us make the following cases:

Case (i): Boy borrows Mathematics Part II, then he borrows Mathematics Part I also.

So the number of possible choices is .

Case (ii) Boy does not borrow Mathematics Part II, then the number of possible choices is.

Hence, the total number of possible choices is .

Question 12:

Find the number of permutations of n different things taken r at a time such that two specific things occur together.

Answer:

A bundle of 2 specific things can be put in r places in ways (Why?) and things in the bundle can be arranged themselves into ways. Now things will be arranged in places in ways.

Thus, using the fundamental principle of counting, the required number of permutations will be

Objective Type Questions

Choose the correct answer out of four options given against each of the following Examples (M.C.Q.).

Question 13:

There are four bus routes between A and B; and three bus routes between B and C. A man can travel round-trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?

(A)

(B)

(C)

(D)

Answer:

(A) Is the correct answer. In the following figure:

Question 13-Four bus and Three bus Routes

Four Bus and Three Bus Routes

there are 4 bus routes from A to B and 3 routes from B to C. Therefore, there are ways to go from A to C. It is round trip so the man will travel back from C to A via B. It is restricted that man can-not use same bus routes from C to B and B to A more than once. Thus, there are routes for return journey. Therefore, the required number of ways .

Question 14:

In how many ways a committee consisting of 3 men and 2 women, can be chosen from 7 men and 5 women?

(A)

(B)

(C)

(D)

Answer:

(B) Is the correct choice.

Out of 7 men, 3 men can be chosen in 7C3 ways and out of 5 women, 2 women can be chosen in ways. Hence, the committee can be chosen in ways.

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