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 integral solutions for the equation $x+y+z+t=20,$ where $x,y,z,t\ge -1,$ is

How many divisors of $21600$ are divisible by $10$ but not by $15?$

The total number of ways in which $5$ balls of different colours can be distributed among $3$ persons so that each person gets atleast one ball is

Total number of different ways in which $7$ different prizes be awarded to $5$ students in such a way that each student has at least one prize is $P\mathit{.}$ Find the value of $\frac{\sqrt{P+100}}{2}$

Let $A$ represents the number of factors of $1800$ and $B$ represents the number of ways in which $1800$ can be resolved into two coprime factors. Find the value of $A-B^2.$

In how many ways can $12$ different things be equally distributed among $4$ persons? If they are divided into $4$ groups instead of giving away to $4$ persons, what will be the number of ways?

In how many different ways can $15$ distinct objects be divided

into $3$ groups of $5$ objects each;

In how many different ways can $15$ distinct objects be divided
in $3$ groups of $4,5$ and $6$ objects;

A person has in his bag $14$ notes of $\mathrm{Rs}.100$ each, $9$ notes of $\mathrm{Rs}. 50$ each, $4$ notes of $\mathrm{Rs}. 20$ each and $7$ notes of $\mathrm{Rs}. 10$ each. In how many different ways can be donate to a poor fund?

There are $4$ ten-rupee, $5$ five-rupee and $3$ two-rupee notes. How many selections of at least one ten-rupee and one two-rupee notes are possible?

Find the number of factors (except $1$) of $420$, excepting the number itself.

In how many different ways the number $1440$ be expressed as the product of two factors (considering $1$ as a factor)?

How many different factors (except 1) are there of the numbers 

$1155$

How many different factors (except 1) are there of the numbers 

$113400$

How many different factors (except 1) are there of the numbers 
$546$ 

A Committee of $12$ is to be formed from $9$ women and $8$ men. In how many ways this can be done if at least $5$ women have to be included in a committee? In how many of these committees

The women are in majority?

If the number of ways in which four distinct balls can be put into two identical boxes such that no box remains empty is equal to $k$, then $k$ is

Let $P$ denotes the ways in which persons $A, B \& C$ will get $1, 2, 5$ objects respectively out of $10$ distinct objects. If $P=\lambda \left(\frac{10!}{1! 2! 5!}\right)$, then $4\lambda =$