Permutations & Combinations · JEE Main previous year questions

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ________.

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then $\text{b}+3\text{ g}$ is equal to _________.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is ${\tan }^{-1}\frac{3}{4}$. Water is poured in it at a constant rate of $6$ cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is $4$ meters, is ____.

The number of 5-digit natural numbers, such that the product of their digits is 36 , is __________.

The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.

The number of ways to distribute 30 identical candies among four children $C_1,C_2,C_3$ and $C_4$ so that $C_2$ receives at least 4 and at most 7 candies, $C_3$ receives at least 2 and at most 6 candies, is equal to :

1. A.$205$
2. B.$615$
3. C.$510$
4. D.$430$

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _____________.

There are ten boys $B_1,B_2,.......,B_{10}$ and five girls $G_1,G_2,.......,G_5$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_1$ and $B_2$ together should not be the members of a group, is ____.

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ___________.

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to _______________.

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

Water is being filled at the rate of $1{\text{ cm}}^3\text{ / sec}$ in a right circular conical vessel (vertex downwards) of height $35\text{ cm}$ and diameter $14\text{ cm}$. When the height of the water level is $10\text{ cm}$, the rate (in ${\text{cm}}^2\text{ / sec}$) at which the wet conical surface area of the vessel increases is

1. A.$5$
2. B.$\frac{\sqrt{21}}{5}$
3. C.$\frac{\sqrt{26}}{5}$
4. D.$\frac{\sqrt{26}}{10}$

The total number of three-digit numbers, with one digit repeated exactly two times, is ______________.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____________.

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is _____________.

A possible value of 'x', for which the ninth term in the expansion of ${\left\{3^{{\log }_3\sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right){\log }_3(5^{x-1}+1)}\right\}}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right){\log }_3(5^{x-1}+1)}$ is equal to 180, is :

1. A.0
2. B.$-1$
3. C.2
4. D.1

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100 k, then k is equal to _____________.

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ______________.

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :

1. A.26664
2. B.122664
3. C.122234
4. D.22264

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is :

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

1. A.35
2. B.42
3. C.82
4. D.77

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after $\frac{k}{{\log }_e\left(\frac{6}{5}\right)}$ hours, then ${\left(\frac{k}{{\log }_{e}2}\right)}^2$ is equal to :

1. A.16
2. B.8
3. C.2
4. D.4

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.

The total number of two digit numbers 'n', such that $3^n+7^n$ is a multiple of 10, is ______ .