Division and Distribution of Objects

How many different ways can $15$ Candy bars be distributed among Ram, Shyam, Ghanshyam and Balram, if Ram can not have more than $5$ candy bars and Shyam must have at least two. Assume all Candy bars to be alike.

Find the number of positive integers that are divisors of at least one of the numbers ${\text{10}}^{10}{\text{; 18}}^{11}\text{;}{15}^7$.

The number of different garlands, that can be formed using $3$ flowers of one kind and $3$ flowers of other kind, is

Number of ways in which $32$ identical blankets can be distributed among $4$ persons so that each of them gets number of blankets which are divisible by $2$ but not by $4$, is

The number of non-negative integral solution of the inequality $A+B+C+D+E\le 2022$ is equal to 

If ${\lambda }_1, {\lambda }_2, {\lambda }_3$ are odd natural numbers, then the number of solutions of ${\lambda }_1+{\lambda }_2+{\lambda }_3=203$ is 

If the integers from $1$ to $2021$ are written as a single integer like $123\dots 91011\dots 20202021$, then the ${2021}^{\mathrm{st}}$ digit (counted from the left) in the resulting number is

The number of $4$ digit natural numbers, such that the sum of the digits of each number is equal to $9$, is

The number of ways in which $10$ apples, $5$ oranges and $5$ mangoes can be distributed among three persons, each receiving none, one or more is :
(assume fruits of the same species to be identical)

Let $N$ be the least positive integer such that whenever a non-zero digit $C$ is written after the last digit of $N$, the resulting number is divisible by $C$. The sum of the digits of $N$ is

For any natural number $n$, $\left(15\times 5^{2n}\right)+\left(2\times 2^{3n}\right)$ is divisible by

In How many ways can $17$ persons depart from airbase in $2$ MIG and $3$ Mirage Fighter jet, given that $2$ particular persons Abhinandan and Jayant depart by same MIG, ($4$ persons can sit in a MIG and $3$ persons can sit in a Mirage)

The number of ways in which $10$ boys can be divided into $2$ groups of $5,$ such that two tallest boys are in two different groups, is equal to

Find the number of ways of selecting four letters from the letters of the word $\text{'}BARBOSSA\text{'}$

For $N=12600$, the number of even divisors is

Five types of ice-creams are available in a shop. A child buys six ice-creams.
Statement-1: The number of different ways the child can buy the six ice-creams is ${{}^{10}C}_5$.
Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 A\text{'}s$ and $4 B\text{'}s$ in a row.

Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

There are $(p+q)$ different books on different topics in Mathematics, where $p\ne q$. If $L$ denotes the number of ways in which these books are distributed between two students $X$ and $Y$ such that $X$ gets $p$ books and $Y$ gets $q$ books, $M$ denotes the number of ways in which these books are distributed between two students $X$ and $Y$ such that one of them gets $p$ books and another gets $q$ books, and $N$ denotes the number of ways in which these books are divided into two groups of $p$ books and $q$ books, then:

Number of ways in which $16$ players can be divided into four equal groups, is :

The number of non-negative integral solutions of $2x+y+z=21$ are

Three schools send $2, 4$ and $6$ students, respectively, to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both from the same school. The number of ways, the students can be accommodated in the rooms is