Fundamental Principle of Counting

Q.

In a survey of 25 students, it was found that 12 have taken physics, 11 have taken chemistry and 15 have taken mathematics; 4 have taken physics and chemistry; 9 have taken physics and mathematics; 5 have taken chemistry and mathematics while 3 have taken all the three subjects.
Find the number of students who have taken

(i) physics only;
(ii) chemistry only;
(iii) mathematics only;
(iv) physics and chemistry but not mathematics;
(v) physics and mathematics but not chemistry;
(vi) only one of the subjects;
(vii) at least one of the three subjects;
(viii) none of the three subjects.

Q.

Numbers greater than $1000$ but not greater than $4000$ which can be formed with the digits $0,1,2,3,4$ (repetition of digits is allowed), are

1. $350$

2. $375$

3. $450$

4. $576$

Q.

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

Q.

In how many ways can 5 different balls be distributed among three boxes?

1. 3!

2. 5³

3. 5!

4. 3⁵

Q.

There are 3 white, 4 red, and 1 blue marbles in a bag. They are drawn one by one and arranged in a row. Assuming that all 8 marbles are drawn, determine the number of different arrangements if marbles of same colour are indistinguishable.

1. $296$

2. $280$

3. $240$

4. $56$

Q.

Write the following intervals in set-builder form :

(i) (-3, 0) (ii) [6, 12]

(iii) [-23, 5]

Q.

There are 4 routes between Delhi and Patna. In how many different ways can a man go from Delhi to Patna and return, if for returning

(i) any of the routes is taken;

(ii) the same route is taken ;

(iii) the same route is not taken ?

Q.

How many three digit odd numbers can be formed by using the digits 1, 2, 3, 4, 5, 6 if the repetition of digit is not allowed:

1. 216

2. 120

3. 60

4. None of these

Q.

How many numbers are there between 100 and 1000 such that every digit is either 2 or 9?

Q.

How do you find the sum of $1$ to $100$?

Q. 30. 25paise and 50 paise total 9 coins how to select 9 coins to make rupee a even number

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

Q. There are 5 multiple choice questions in test. If the first three questions have 4 choices each and the next two have 5 choices each, the number of answers possible is

1. 1500
2. 1600
3. 1800
4. 1700

Q.

The sum of the integers lying between 1 and 100 and divisible by 3 or 5 or 7 is

1. $2738$
2. $3368$
3. $93468$
4. $2838$

Q.

A room has 8 doors. In how many ways, a man can enter in the room through one door and exit through a different door?

1. 8

2. 5040

3. 56

4. 40320

Q.

A coin is tossed 6 times, in how many throws can 4 heads and 2 tails be obtained?

1. 24

2. 10

3. 15

4. 18

Q. 27.A party of 6 is to be formed from 10 boys and 7 girls so as to include 3 boys and 3 girls. in how many different ways can be formed if two particular girls refuse to join the same party.

Q.

The number of positive integers greater than 6000 but not more than 7000, divisible by 5, using the digits 0, 1, 2, ........9, if any digit can be used any numbers of times, is

1. 200
   

2. 112
   

3. 119
   

4. None of these

Q.

How many 6 digit telephone numbers can be constructed with the digits 0 to 9 , if each number starts with 23 and if repetition of digits is not allowed.

1. 720

2. 1680

3. 10000

4. 5040

Q.

The number of numbers lying between 99 and 1000 which can be formed using the digits 2, 3, 7, 0, 6, 8 when no digit is being repeated is

1. 102
2. 120
3. 100
4. None of these

Q. ntOut of 20 games of chess played by two players A and B, A won 12, B won 4 and 4 ended in a tie. In a tournament of 3 games find the probality that i) B wins all 3, ii) B wins at least 1, iii) 2 games end in a tie.n

Q. A water tank is filled by three pipes with uniformflow. The first two pipes operating simultaneously, fill the tank in the same time during which the tank isfilled by the third pipe alone. The first pipe fills the tank six hours slower than the second pipe andeight hours slower than the third pipe. The time takenby the second pipe to fill the tank is what?

Q.

The number of distinct rational numbers of the form p/q, where p, q $ϵ$ {1, 2, 3, 4, 5, 6} is

1. 23
2. 36
3. 32
4. 28

Q. a certain number of students of a school participated in a chess tournament of their annual sports meet. each player played one game against each of the other player it was found that in 66 games both the players were girls and in 240 games one was a girl and other was boy the number of games in which both the players were boys is

Q.

In a certain test ${\mathrm{a}}_{\mathrm{i}}$ students gave wrong answers to at least $\mathrm{i}$ questions, where $\mathrm{i}=1,2,3...\mathrm{k}$ No student gave more than $\mathrm{k}$ wrong answers. The total number of wrong answers is

1. ${\mathrm{a}}_1+{\mathrm{a}}_2...+{\mathrm{a}}_{\mathrm{k}}$

2. ${\mathrm{a}}_1+2{\mathrm{a}}_2...+{\mathrm{ka}}_{\mathrm{k}}$

3. ${\mathrm{a}}_1+{\mathrm{a}}_2...+{\mathrm{a}}_{\mathrm{k}}-\mathrm{k}$

4. $\frac{\mathrm{k}}{2\left(\mathrm{k}+1\right)}$

Q. The number of six letter words that can be formed using the letters of the word "ASSIST" in which S's alternate with other letters is
(a) 12
(b) 24
(c) 18
(d) none of these.

Q.

How many three digit numbers are there with distinct digit with each digit odd?

1. 60

2. 70

3. 80

4. 50

Q. Number of $4$ digit numbers that can be formed using the digits $0,1,2,3,4,5$ which are divisible by $6$ when repetition of digits is not allowed are

1. $24$
2. $52$
3. $72$
4. $48$

Q. a postman has to dilever five letters to five different house. mischievously, he post one letter through each door without looking to see if it is the correct address. in how many different ways could he do this so that exactly two of the five houses receieve the correct letter

Q. The number of ways in which $10$ candidates $A_1,A_2,A_3,\dots ,A_{10}$ can be ranked so that $A_1$ is always above $A_2,$ is

1. $10!$
2. $\frac{10!}{2}$
3. $9!$
4. None of these