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

Let the coefficient of $x^{1009}$ in the polynomial equation $P\left(x\right)=\prod _{r=0}^{1009}\left(x+{}^{2019}C_r\right)$ is  in the form $m^n$ and sum of divisors of $n$ is the product of numbers $a$ and $b$ where, $a$ is composite number and $b$ is a three-digit prime number, then

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 $50$.

Let '$x$' be the number of ways in which $7$ identical balls can be coloured using $5$ different colours, no ball is supposed to be coloured with more than one colour and all the $5$ colours are to be used & '$y$' be the number of ways in which the aforesaid work is done, when the balls are numbered.
Let "$a$" be the number of ways in which $7$ distinct balls can be distributed into $5$ identical boxes so that no box is empty.
And "$b$" be the number of ways in which $7$ distinct balls can be distributed into $5$ distinct boxes so that no box is empty, then which of the following is/are true?

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

Consider a cube of side length $9$ units. Let $\left(x,y,z\right)$ be coordinates of points on or inside the cube such that $x,y,z\in I$ and $0\le x,y,z\le 9$. If total number of ways of selecting two distinct points among these such that their mid-point is also having integral coordinates is $N$, then which of the following is/are TRUE?

Let $n$ be number of distinct objects, $k$ be number of like boxes and let $P\left(n,k\right)$ be the number of ways of putting all the $n$ objects into the boxes so that no box is empty then which is/are correct ?

Let $f\left(x\right)=x^{13}+2x^{12}+3x^{11}+\dots \dots +13x+14$ and $\alpha =\cos \frac{2\pi }{15}+i\sin \frac{2\pi }{15},\left(i=\sqrt{-1}\right)$. If $N=f\left(\alpha \right)·f\left({\alpha }^2\right)·f\left({\alpha }^3\right)\dots .·f\left({\alpha }^{14}\right)$ then which of the following is/are INCORRECT?

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 
$546$