**Chapter 3 Permutations and Combination Ex 3.1**

## Chapter 3 Permutations and Combination Ex 3.1

**Question 1.A teacher wants to select the class monitor in a class of 30 boys and 20 girls. In how many ways can the monitor be selected if the monitor must be a girl?Solution:**

There are 30 boys and 20 girls.

A teacher can select any boy as a class monitor from 30 boys in 30 different ways and he can select any girl as a class monitor from 20 girls in 20 different ways.

∴ by the fundamental principle of addition, the total number of ways a teacher can select a class monitor = 30 + 20 = 50

Hence, there are 50 different ways to select a class monitor.

**Question 2.A Signal is generated from 2 flags by putting one flag above the other. If 4 flags of different colours are available, how many different signals can be generated?Solution:**

A signal is generated from 2 flags and there are 4 flags of different colours available.

∴ 1st flag can be any one of the available 4 flags.

∴ It can be selected in 4 ways.

Now, 2nd flag is to be selected for which 3 flags are available for a different signal.

∴ 2nd flag can be anyone from these 3 flags.

∴ It can be selected in 3 ways.

∴ By using the fundamental principle of multiplication, total no. of ways a signal can be generated = 4 × 3 = 12

∴ 12 different signals can be generated.

**Question 3.How many two-letter words can be formed using letters from the word SPACE, when repetition of letters (i) is allowed (ii) is not allowedSolution:**
A two-letter word is to be formed out of the letters of the word SPACE.

(i) When repetition of the letters is allowed 1st letter can be selected in 5 ways.

2nd letter can be selected in 5 ways.

∴ By using the fundamental principle of multiplication, the total number of 2-letter words = 5 × 5 = 25

(ii) When repetition of the letters is not allowed 1st letter can be selected in 5 ways.

2nd letter can be selected in 4 ways.

∴ By using the fundamental principle of multiplication, the total number of 2-letter words = 5 × 4 = 20

**Question 4.How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits (i) are allowed (ii) are not allowed?Solution:**
A three-digit number is to be formed from the digits 0, 1, 3, 5, 6.

(i) When repetition of digits is allowed,

100’s place digit should be a non-zero number.

Hence, it can be anyone from digits 1, 3, 5,6.

∴ 100’s place digit can be selected in 4 ways.

10’s and unit’s place digit can be zero and digits can be repeated.

∴ 10’s place digit can be selected in 5 ways and the unit’s place digit can be selected in 5 ways.

∴ By using the fundamental principle of multiplication, the total number of three-digit numbers = 4 × 5 × 5 = 100

(ii) When repetition of digits is not allowed,

100’s place digit should be a non-zero number.

Hence, it can be anyone from digits 1, 3, 5, 6.

∴ 100’s place digit can be selected in 4 ways.

10’s and unit’s place digit can be zero and digits can’t be repeated.

∴ 10’s place digit can be selected in 4 ways and the unit’s place digit can be selected in 3 ways.

∴ By using the fundamental principle of multiplication, the total number of two-digit numbers = 4 × 4 × 3 = 48

**Question 5.How many three-digit numbers can be formed using the digits 2, 3, 4, 5, 6 if digits can be repeated?Solution:**
A 3-digit number is to be formed from the digits 2, 3, 4, 5, 6 where digits can be repeated.

∴ Unit’s place digit can be selected in 5 ways.

10’s place digit can be selected in 5 ways.

100’s place digit can be selected in 5 ways.

∴ By using fundamental principle of multiplication, total number of 3-digit numbers = 5 × 5 × 5 = 125

**Question 6.A letter lock contains 3 rings and each ring contains 5 letters. Determine the maximum number of false trails that can be made before the lock is opened.Solution:**
A letter lock has 3 rings, each ring containing 5 different letters.

∴ A letter from each ring can be selected in 5 ways.

∴ By using the fundamental principle of multiplication, a total number of trials that can be made = 5 × 5 × 5 = 125.

Out of these 124 wrong attempts are made and in the 125th attempt, the lock gets opened.

∴ A maximum number of false trials = 124.

**Question 7.In a test, 5 questions are of the form ‘state, true or false. No student has got all answers correct. Also, the answer of every student is different. Find the number of students who appeared for the test.Solution:**

Every question can be answered in 2 ways. (True or False)

∴ By using the fundamental principle of multiplication, the total number of set of answers possible = 2 × 2 × 2 × 2 × 2 = 32.

Since One of them is the case where all questions are answered correctly,

The number of wrong answers = 32 – 1 = 31.

Since no student has answered all the questions correctly, the number of students who appeared for the test are 31.

**Question 8.How many numbers between 100 and 1000 have 4 in the unit’s place?Solution:**
Numbers between 100 and 1000 are 3-digit numbers.

A 3-digit number is to be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where the unit place digit is 4.

Unit’s place digit is 4.

∴ it can be selected in 1 way only.

10’s place digit can be selected in 10 ways.

For a 3-digit number, 100’s place digit should be a non-zero number.

∴ 100’s place digit can be selected in 9 ways.

∴ By using the fundamental principle of multiplication,

total numbers between 100 and 1000 which have 4 in the units place = 1 × 10 × 9 = 90.

**Question 9.How many numbers between 100 and 1000 have the digit 7 exactly once?Solution:**
Numbers between 100 and 1000 are 3-digit numbers.

A 3-digit number is to be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where exactly one of the digits is 7.

When 7 is in unit’s place:

Unit’s place digit is 7.

∴ it can be selected in 1 way only.

10’s place digit can be selected in 9 ways.

100’s place digit can be selected in 8 ways.

Total numbers which have 7 in unit’s place = 1 × 9 × 8 = 72.

When 7 is in 10’s place:

Unit’s place digit can be selected in 9 ways.

10’s place digit is 7.

∴ it can be selected in 1 way only.

100’s place digit can be selected in 8 ways.

∴ A total number of numbers which have 7 in 10’s place = 9 × 1 × 8 = 72.

When 7 is in 100’s place:

Unit’s place digit can be selected in 9 ways.

10’s place digit can be selected in 9 ways.

100’s place digit is 7.

∴ it can be selected in 1 way only.

∴ Total numbers which have 7 in 100’s place = 9 × 9 × 1 = 81.

∴ Total numbers between 100 and 1000 having digit 7 exactly once = 72 + 72 + 81 = 225

**Question 10.How many four-digit numbers will not exceed 7432 if they are formed using the digits 2, 3, 4, 7 without repetition?Solution:**

Between any set of digits, the greatest number is possible when digits are arranged in descending order.

∴ 7432 is the greatest number, formed from the digits 2, 3, 4, 7.

Since a 4-digit number is to be formed from the digits 2, 3, 4, 7, where repetition of the digit is not allowed,

1000’s place digit can be selected in 4 ways,

100’s place digit can be selected in 3 ways,

10’s place digit can be selected in 2 ways,

Unit’s place digit can be selected in 1 way.

∴ Total number of numbers not exceeding 7432 that can be formed with the digits 2, 3, 4, 7 = Total number of four-digit numbers possible from the digits 2, 3, 4, 7

= 4 × 3 × 2 × 1

= 24

**Question 11.If numbers are formed using digits 2, 3, 4, 5, 6 without repetition, how many of them will exceed 400?Solution:**
Case I: Number of three-digit numbers formed from 2, 3, 4, 5, 6, greater than 400.

100’s place can be filled by any one of the numbers 4, 5, 6.

100’s place digit can be selected in 3 ways.

Since repetition is not allowed, 10’s place can be filled by any one of the remaining four numbers.

∴ 10’s place digit can be selected in 4 ways.

Unit’s place digit can be selected in 3 ways.

∴ Total number of three-digit numbers formed = 3 × 4 × 3 = 36

Case II: Number of four-digit numbers formed from 2, 3, 4, 5, 6.

Since repetition of digits is not allowed,

1000’s place digit can be selected in 5 ways.

100’s place digit can be selected in 4 ways.

10’s place digit can be selected in 3 ways.

Unit’s place digit can be selected in 2 ways.

∴ Total number of four-digit numbers formed = 5 × 4 × 3 × 2 = 120

Case III: Number of five-digit numbers formed from 2, 3, 4, 5, 6.

Similarly, since repetition of digits is not allowed,

Total number of five digit numbers formed = 5 × 4 × 3 × 2 × 1 = 120.

∴ Total number of numbers that exceed 400 = 36 + 120 + 120 = 276

**Question 12.How many numbers formed with the digits 0, 1, 2, 5, 7, 8 will fall between 13 and 1000 if digits can be repeated?Solution:**
Case I: 2-digit numbers more than 13, less than 20, formed from the digits 0, 1, 2, 5, 7, 8.

10’s place digit is 1.

∴ it can be selected in 1 way only.

Unit’s place can be filled by any one of the numbers 5, 7, 8.

∴ Unit’s place digit can be selected in 3 ways.

∴ Total number of such numbers = 1 × 3 = 3.

Case II: 2-digit numbers more than 20 formed from 0, 1, 2, 5, 7, 8.

10’s place can be filled by any one of the numbers 2, 5, 7, 8.

∴ 10’s place digit can be selected in 4 ways.

Since repetition is allowed, the unit’s place can be filled by one of the remaining 6 digits.

∴ Unit’s place digit can be selected in 6 ways.

∴ Total number of such numbers = 4 × 6 = 24.

Case III: 3-digit numbers formed from 0, 1, 2, 5, 7, 8.

Similarly, since repetition of digits is allowed, the total number of such numbers = 5 × 6 × 6 = 180.

All cases are mutually exclusive.

∴ Total number of required numbers = 3 + 24 + 180 = 207

**Question 13.A school has three gates and four staircases from the first floor to the second floor. How many ways does a student have to go from outside the school to his classroom on the second floor?Solution:**

A student can go inside the school from outside in 3 ways and from the first floor to the second floor in 4 ways.

∴ A number of ways to choose gates = 3.

The number of ways to choose a staircase = 4.

By using the fundamental principle of multiplication,

number of ways in which a student has to go from outside the school to his classroom = 4 × 3 = 12

**Question 14.How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?Solution:**
Here, repetition of digits is not allowed.

For a number to be divisible by 5,

unit’s place digit should be 0 or 5.

Case I: when unit’s place is 0

Unit’s place digit can be selected in 1 way.

10’s place digit can be selected in 5 ways.

100’s place digit can be selected in 4 ways.

1000’s place digit can be selected in 3 ways.

10000’s place digit can be selected in 2 ways.

∴ Total number of numbers = 1 × 5 × 4 × 3 × 2 = 120.

Case II: when the unit’s place is 5

Unit’s place digit can be selected in 1 way.

10000’s place should be a non-zero number.

∴ It can be selected in 4 ways.

1000’s place digit can be selected in 4 ways.

100’s place digit can be selected in 3 ways.

10’s place digit can be selected in 2 ways.

∴ Total number of numbers = 1 × 4 × 4 × 3 × 2 = 96

∴ Total number of required numbers = 120 + 96 = 216