# Exercise 7.3(**Permutations and Combinations)**

### Chapter 7 Exercise 7.3 ncert solutions maths class 11

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

**Solution:** Total number of digit

Number of places where the digit can fill

If repetition of digit not allowed,

Then number of Permutations

.

**Question 2: How many 4-digit numbers are there with no digit repeated?**

**Solution: **Now, we will have 4 places where 4 digits are to be put.

So, at thousand’s place = There are 9 ways, as 0 cannot be at thousand’s place

At hundredth’s place = There are 9 digits to be filled, as 1 digit is already taken = 9 ways

At ten’s place = There are now 8 digits to be filled, as 2 digits are already taken = 8 ways

At unit’s place = There are 7 digits that can be filled = 7 ways

**The total number of ways to fill the four places = 9 × 9 × 8 × 7 = 4536 ways**

**So, a total of 4536 four-digit numbers can be there with no digits repeated.**

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

**Solution: **An even number means that the last digit should be even.

Number of possible digits at one’s place = 3 (2, 4 and 6)

Number of permutations

One of the digits is taken at one’s place; the number of possible digits available = 5

Number of permutations

Therefore, the total number of permutations =3 × 20=60

### OR

3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7,** = 3 × 5 × 4 = 60**

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

**Solution: **An even number means that the last digit should be even.

Number of possible digits at one’s place = 3 (2 and 4 )

Number of permutations

One of the digits is taken at one’s place; the number of possible digits available = 4

Number of permutations

Therefore, the total number of permutations =2 × 24=48

### OR

The number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 if no digit is repeated = 4 × 3 × 2 × 2 = 48

**Question 5:** **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?**

**Solution: **Total number of people in committee

Number of positions to be filled

Number of permutations

### OR

Number of way of selection of chairman and a vice chairman from 8 person = 8 × 7 = 56

**Question 6:** **Find n if ^{n-1}P_{3}: ^{n}P_{3} = 1: 9.**

**Solution: Given that**

**Question 7: Find r if**

**(i) ^{5}P_{r} = 2^{6}P_{r-1} **

**(ii) ^{5}P_{r} = ^{6}P_{r-1}**

**Solution: (i) Given that**

^{5}P_{r} = 2^{6}P_{r-1}

or

But r = 10 is rejected, r can not be greater than 5,

Therefore

**(ii)** `Given that,

** ^{5}P_{r} = ^{6}P_{r-1}**

or

But r = 9 is rejected, r can not be greater than 5,

Therefore

**Question 8: How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?**

**Solution: **Total number of different letters in EQUATION = 8

Number of letters to be used to form a word = 8

Number of permutations

= 8 × 7 × 6 × 5 × 4 × 3× 2×1

**= 40320**

**Question 9: 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 the first letter is a vowel**

**Solution: (i)** Number of letters to be used =4

Number of permutations

**(ii)** Number of letters to be used = 6

Number of permutations

**(iii)** Number of vowels in MONDAY = 2 (O and A)

⇒ Number of permutations in vowel

Now, the remaining places = 5

Remaining letters to be used =5

Number of permutations

Therefore, the total number of permutations = 2 × 120 =240

### OR

**all letters are used, but the first letter is a vowel = 2×5×4×3×2×1 = 240**

**Question 10: In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?**

**Solution: **Total number of letters in MISSISSIPPI = 11

Letter number of occurrence

M → 1

I → 4

S → 4

P → 2

Number of permutations

We take that 4 I’s come together, and they are treated as 1 letter.

∴ Total number of letters=11 – 4 + 1 = 8

⇒ Number of permutations

Therefore, the total number of permutations where four I’s don’t come together = 34650-840=33810

**Question 11:** **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**

**(iii) There are always 4 letters between P and S**

**Solution:** **(i)** Total number of letters in PERMUTATIONS =12

The only repeated letter is T = 2times

The first and last letters of the word are fixed as P and S, respectively.

Number of letters remaining =12 – 2 = 10

⇒ Number of permutations

**(ii)** Number of vowels in PERMUTATIONS = 5 (E, U, A, I, O)

Now, we consider all the vowels together as one.

Number of permutations of vowels = 120

Now total number of letters = 12 – 5 + 1= 8

⇒ Number of permutations

Therefore, total number of permutations = 120 × 20160 = 2419200

**(iii)** Number of places is as 1 2 3 4 5 6 7 8 9 10 11 12

There should always be 4 letters between P and S.

Possible places of P and S are 1 and 6, 2 and 7, 3 and 8, 4 and 9, 5 and 10, 6 and 11, 7 and 12

Possible ways =7

Also, P and S can be interchanged.

No. of permutations =2 × 7 =14

The remaining 10 places can be filled with 10 remaining letters,

∴ No. of permutations

Therefore, the total number of permutations = 14 × 1814400 =25401600

### Chapter 7 Permutation and Combination Ncert math solution

Exercise 7.1 Permutation and Combination Ncert math solution

Exercise 7.2 Permutation and Combination Ncert math solution

Exercise 7.3 Permutation and Combination Ncert math solution