Division and Distribution of Objects

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

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 $9$ different objects be distributed in $3$ persons in such a way that they get two, three and four objects?

In how many ways can $10$ different objects be distributed in $3$ persons in such a way that they get two, three and five objects?

The number of positive divisors of the number $79200$ which are not divisible by $4$ is

A positive integer $k$, is perfect if the sum of its positive divisors equals $2k$. If $n$, is a perfect number, then the sum of the reciprocal of its positive divisors

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

Find the product of all the factors of the given number.

$16$

Find the product of all the factors of the given number.

$12$

In how many ways $3600$ can be resolved as a product of two different factors.

In how many ways $2500$ can be resolved as a product of two different factors.

In how many ways the number $10800$ can be resolved as a product of two different factors.

In how many ways $8100$ can be resolved as a product of two different factors.

In how many ways can $38808$ be expressed as a product of two co primes?

In how many ways can $1200$ be expressed as a product of two co primes?

A question paper has two parts, Part $A$ and Part $B$, each containing $10$ questions. If a student has to choose $8$ from Part $A$ and $5$ from Part $B$, in how many ways can he choose the questions?

Three gentlemen and three ladies are candidates for two vacancies. A voter has to vote for two candidates. In how many ways can one cast his vote?