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 $……$

Team $'A'$ consists of $7$ boys and $n$ girls and Team $'B'$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to

In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote

The value of $r$ for which $^{20}{C_r}^{20}{C_0}{ + ^{20}}{C_{r – 1}}^{20}{C_1}{ + ^{20}}{C_{r – 2}}^{20}{C_2} + …{ + ^{20}}{C_0}^{20}{C_r}$ is maximum is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these

cards are of the same colour?