Amit M Agarwal Solutions for Chapter: Permutation and Combination, Exercise 6: Work Book Exercise 6.6

The number of non-negative integral solutions of $x+y+z\le n,$ where $n\in N$ is

If $n$ objects are arranged in a row, then the number of ways of selecting three of these objects, so that no two of them are next to each other, is

There are three piles of identical red, blue and green balls and each pile contains at least $10$ balls. The number of ways of selecting $10$ balls, if twice as many red balls as green balls are to be selected, is

Two classrooms $A$ and $B$ having capacity of $25$ and $\left(n-25\right)$ seats, respectively. Let $A_n$ denotes the number of possible seating arrangements of room $A$, when $n$ students are to be seated in these rooms, starting from room A which is to be filled up full to its capacity. If $A_n-A_{n-1}=25!\left({}^{49}\mathrm{C}_{25}\right)$, then $n$ equals

Between two junction stations $A$ and $B$ there are $12$ intermediate stations. The number of ways in which a train can be made to stop at $4$ of these stations, so that no two of these halting stations are consecutive, is

The sum of the divisors of $2^5·3^4·5^2$ is

The number of ways arranging $m$ positive and $n$ negative signs in a row, so that no two negative signs are together, is

$\left(n<m+1\right)$

If $a_j, i=1, 2, 3, 4$ is four real numbers of the same sign, then the minimum value of $\Sigma \frac{a_i}{a_j}$ $i, j\in \left\{1,2,3,4\right\},i\ne j,$ is