# NCERT Solutions for Class 11 Maths Maths Chapter 7

## Permutations and Combinations Class 11

### Exercise 7.1 : Solutions of Questions on Page Number : 138

Q1 :

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that

(i) repetition of the digits is allowed?

(ii) repetition of the digits is not allowed?

(i)

There will be as many ways as there are ways of filling 3 vacant places

in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filled in by any of the given five digits. Similarly, tens and hundreds digits can be filled in by any of the given five digits.

Thus, by the multiplication principle, the number of ways in which three-digit numbers can be formed from the given digits is 5 x 5 x 5 = 125

(ii)

In this case, repetition of digits is not allowed. Here, if units place is filled in first, then it can be filled by any of the given five digits. Therefore, the number of ways of filling the units place of the three-digit number is 5.

Then, the tens place can be filled with any of the remaining four digits and the hundreds place can be filled with any of the remaining three digits.

Thus, by the multiplication principle, the number of ways in which three-digit numbers can be formed without repeating the given digits is 5 x 4 x 3 = 60

Q2 :

How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

There will be as many ways as there are ways of filling 3 vacant places

in succession by the given six digits. In this case, the units place can be filled by 2 or 4 or 6 only i.e., the units place can be filled in 3 ways. The tens place can be filled by any of the 6 digits in 6 different ways and also the hundreds place can be filled by any of the 6 digits in 6 different ways, as the digits can be repeated.

Therefore, by multiplication principle, the required number of three digit even numbers is 3 x 6 x 6 = 108

Q3 :

How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Q4 :

How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

Q5 :

A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Q6 :

Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

### Exercise 7.2 : Solutions of Questions on Page Number : 140

Q1 :

Evaluate

(i) 8! (ii) 4! - 3!

Q2 :

Is 3! + 4! = 7!?

Q3 :

Compute

Q4 :

If , find x.

Q5 :

Evaluate, when

(i) n = 6, r = 2 (ii) n = 9, r = 5

### Exercise 7.3 : Solutions of Questions on Page Number : 148

Q1 :

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

Q2 :

How many 4-digit numbers are there with no digit repeated?

Q3 :

How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?

Q4 :

Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

Q5 :

From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?

Q6 :

Find n if

Q7 :

Find r if (i) (ii) .

Q8 :

How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?

Q9 :

How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if

(i) 4 letters are used at a time, (ii) all letters are used at a time,

(iii) all letters are used but first letter is a vowel?

Q10 :

In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together?

Q11 :

In how many ways can the letters of the word PERMUTATIONS be arranged if the

(i) words start with P and end with S, (ii) vowels are all together,

(ii) there are always 4 letters between P and S?

### Exercise 7.4 : Solutions of Questions on Page Number : 153

Q1 :

If, find.

Q2 :

Determine n if

(i) (ii)

Q3 :

How many chords can be drawn through 21 points on a circle?

Q4 :

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Q5 :

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Q6 :

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Q7 :

Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Q8 :

In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Q9 :

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Q10 :

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

### Exercise Miscellaneous : Solutions of Questions on Page Number : 156

Q1 :

How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?

Q2 :

How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

Q3 :

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:

(i) exactly 3 girls? (ii) atleast 3 girls? (iii) atmost 3 girls?

Q4 :

If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

Q5 :

How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?

Q6 :

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

Q7 :

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Q8 :

Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

Q9 :

It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

Q10 :

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

Q11 :

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?