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 ${\text{10}}^{10}{\text{; 18}}^{11}\text{;}{15}^7$.

If $16$ identical pencils are distributed among $4$ children such that each gets at least $3$ pencils. The number of ways of distributing the pencils is

Number of positive integral solutions of $x_1{x}_2{x}_3=30,$ is

The number of triplets $\left(x, y, z\right)$ where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is

Eight persons are to be transported from city $\mathrm{A}$ to city $\mathrm{B}$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

If $x+y+z=17$ and $x,y,z$ are non-negative integers, then find the number of integral solutions.

If $8$ persons have to travel from Point $A$ to Point $B$ in $3$ allotted cars. If a car can carry maximum $3$ persons then find the number of ways they can travel.

The number of ways to distribute $20$ chocolates among three students such that each student will get atleast one chocolate is

Number of positive integral solutions of $2x_1+3x_2+4x_3+5x_4=25$ is

Number of ways in which $10$ persons can go in two boats so that there may be $5$ on each boat, supposing that two particular persons will not go in same boat is

In how many ways can $10$ balls be divided between two boys one receiving two and the other eight balls?

The number of ways of dividing $8$ things into $4$ equal groups is

Consider seven-digit number $x_1, x_2, ...,x_7$ where, $x_1, x_2,....,x_7\ne 0$ having the property that $x_4$ is the greatest digit and digit towards the left and right of $x_4$ are in decreasing order. The total number of such numbers in which all digits are distinct is

Let $x$ be the elements of the set $A=\left\{1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\right\}$ and $x_1,x_2,x_3$ be positive integers and $d$ be the number of integral solutions of $x_1{x}_2{x}_3=x$, then $d$ is

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

How many $6$ digit numbers have sum of digits $50$.

The value of $\sum _{k=1}^{2011}\frac{k-1}{k!(2011-k)!}=\frac{n\left(2^{n+1}\right)+1}{(n+2)!}$. Number of positive integral divisors of $n$ is

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 $4\lambda =$