Fundamental Principle of Counting

Q.

The sum of all three-digit natural numbers which are divisible by $7$ is

1. $25501$
2. $52005$
3. $70336$
4. $84321$

Q.

The smallest $5$ digit number exactly divisible by $41$ is:

Q. Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20.$ The number of such distinct arrangements $(n_1,n_2,n_3,n_4,n_5)$ is ?

1. 7
2. 6
3. 8
4. 4

Q.

The number of $4$ digits numbers which do not contain $4$ different digit is?

1. $2432$

2. $3616$

3. $4210$

4. $4464$

Q.

$4$ note of Rs.$100$ and $5$ notes in which first of Rs. $1$, second of Rs. $2$, Third of Rs. $5$, fourth of Rs $20$ and fifth one of Rs $50$ distributed among $3$ children such that each child receives at least one note of Rs.$100$. The total number of ways of distribution

1. $3\times 5^3$

2. $5^3\times 3$

3. $3^6$

4. None of these

Q.

A four-digit number $abcd$ is divisible by $11$, if $d+b=$ _______ or _______.

Q.

$10$ different letters of English alphabet are given. Out of these letters, words of $5$ letters are formed. How many words are formed when at least one letter is repeated?

1. $69760$

2. $98748$

3. $96747$

4. $97147$

Q. The number of onto functions $f$ from $\{1,2,3,....,20\}$ to $\{1,2,3,...,20\}$ such that $f\left(k\right)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is :

1. $5!\times 6!$
2. $5^6\times 15$
3. $6^5\times \left(15\right)!$
4. $\left(15\right)!\times 6!$

Q.

$5$ digit number combinations.

Q. The number of $5$-digit numbers that can be formed using the digits $0,2,3,6,8,9$ (without repetition) such that it is divisible by $4$ is

1. $196$
2. $144$
3. $216$
4. $204$

Q.

Ten different letters of English alphabet are given. Out of these letters, words of $5$ letters are formed. How many words are formed when at least one letter is repeated?

1. $69760$

2. $98748$

3. $96747$

4. $97147$

Q. A student is constructing a family tree. If he wants to track back through $10$ generations to calculate the total number of ancestors he has, then the total number of ancestors after the ${10}^{th}$ generation is

1. $2046$
2. $1024$
3. $2048$
4. $1023$

Q. A variable name in certain computer language must be either an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to

1. $280$
2. $290$
3. $286$
4. $296$

Q.

The number of ways in which $6$ rings can be worn on the four fingers of one hand is?

1. $4^6$

2. ${}^{6}C_4$

3. $6^4$

4. None of these

Q. There are 5 doors to a lecture room. The number of ways that a student can enter the room and leave it by a different door is

1. 20
2. 16
3. 19
4. 25

Q.

Write the natural numbers from $30$ to $60$. What fraction of them are divisible by $6?$

Q. The general solution of the inequality $-9\le 1-2(x+3)<1$ and $-5\le \frac{x-9}{2}\le 8$ is

1. 
2. 
3. 
4. 

Q. Find the number of $5$ digit numbers using digits $0,1,2,3,4,5$ (without repetition) such that they are divisible by $5$ :

1. $218$
2. $150$
3. $216$
4. $210$

Q. The number of $4$ digited numbers that can be formed using the digits $1,2,5,6,7$ without repetition and that are divisible by $5$ is

Q. Number of positive integral solutions of $abc=30$ is

1. $9$
2. $27$
3. $81$
4. $243$

Q. The number of $4$ digited numbers that can be formed using the digits $1,2,5,6,7$ (without repetition) that are divisible by $25$ is

Q. The number of $4$ digit numbers that can be formed using the digits $1,2,5,6,7$ without repetition and that are divisible by $3$ is

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. If a fair coin is tossed n times, then the total number of possible outcomes are

1. $2^n$
2. $2^n+1$
3. $2^{n+1}$
4. $2^{n-1}$

Q. Number of $4$ digit numbers that can be formed using digits $0,1,2,3,4,5$ which are divisible by $8,$ when each digit is used at most once is

1. $32$
2. $37$
3. $33$
4. $31$

Q. Number of $4$ digit numbers using digits $0,1,2,3,4,5$ which are divisble by $9$, when each digit used at most once

1. $30$
2. $32$
3. $36$
4. $40$

Q. In how many ways 6 letters can be put in 6 addressed envelopes so that only 5 of them will go in the correct envelopes.

1. 1

2. 0

3. 6!

4. 277

Q. The result of $21$ football matches (win, lose, draw) are to be predicted. The number of different forecasts that can contain $19$ wins is

1. $210$
2. $640$
3. $840$
4. $1260$

Q. In how many ways one can post three letters in four letter boxes?

1. $3^4$
2. $3^3\times 4$
3. $4^3$
4. $3\times 4^3$

Q. In a shelf there are $2$ different physics books and $3$ different chemistry books. The number of ways in which a student can select a physics book or chemistry book is:

1. $5$
2. $6$
3. $7$
4. $4$