If $^n{P_r}$=$ 720$.$^n{C_r},$ then $r$ is equal to

A set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to

The number of ways in which thirty five apples can be distributed among $3$ boys so that each can have any number of apples, is

The least value of natural number $n$ satisfying $C(n,\,5) + C(n,\,6)\,\, > C(n + 1,\,5)$ is

How many $6 -$ digit numbers can be formed from the digits, $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated?

There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100 \mathrm{k}$, then $\mathrm{k}$ is equal to $......$