# NCERT Class 11-Math՚s: Chapter – 7 Permutations and Combinations Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 3**:

In how many ways can this diagram be coloured subject to the following two conditions?

(i) Each of the smaller triangle is to be painted with one of three colours: red, blue or green.

(ii) No two adjacent regions have the same colour.

**Answer**:

These conditions are satisfied exactly when we do as follows: First paint the central triangle in any one of the three colours. Next paint the remaining 3 triangles, with any one of the remaining two colours.

By the fundamental principle of counting, this can be done in ways.

**Question 4**:

In how many ways can 5 children be arranged in a line such that (i) two particular children of them are always together (ii) two particular children of them are never together.

**Answer**:

(i) We consider the arrangements by taking 2 particular children together as one and hence the remaining 4 can be arranged in ways. Again two particular children taken together can be arranged in two ways. Therefore, there are total ways of arrangement.

(ii) Among the permutations of children, there are 48 in which two children are together. In the remaining permutations, two particular children are never together.

**Question 5**:

If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the 49^{th} word?

**Answer**:

Starting with letter A, and arranging the other four letters, there are words. These are the first 24 words. Then starting with G, and arranging A, A, I and N in different ways, there are words. Next the 37^{th} word starts with I.

There are again 12 words starting with I. This accounts up to the 48^{th} word. The 49^{th} word is NAAGI.

**Question 6**:

In how many ways 3 mathematics books, 4 history books, 3 chemistry books and 2 biology books can be arranged on a shelf so that all books of the same subjects are together.

**Answer**:

First we take books of a particular subject as one unit. Thus there are 4 units which can be arranged in ways. Now in each of arrangements, mathematics books can be arranged in ways, history books in ways, chemistry books in 3! ways and biology books in ways. Thus the total number of ways .

**Question 7**:

A student has to answer questions, choosing atleast 4 from each of Parts A and B. If there are questions in Part A and 7 in Part B, in how many ways can the student choose questions?

**Answer**:

The possibilities are:

4 from Part A and 6 from Part B

Or 5 from Part A and 5 from Part B

Or 6 from Part A and 4 from Part B.

Therefore, the required number of ways is

## Long Answer Type

**Question 8**:

Suppose *m* men and *n* women are to be seated in a row so that no two women sit together. If , show that the number of ways in which they can be seated is

**Answer**:

Let the men take their seats first. They can be seated in ways as shown in the following figure

From the above figure, we observe, that there are places for *n* women. It is given that and no two women can sit together. Therefore, *n* women can take their seats ways and hence the total number of ways so that no two women sit together is