Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then,the number of ways of choosing $x$ and $y$ such that $x + y$ is divisible by $5$ is $.........$.

The number of even numbers greater than $1000000$ that can be formed using all the digits $1, 2, 0, 2, 4, 2, 4$ is

If the letters of the word $ATTEMPT$ are rearranged randomly,what is the probability that all $T$'s occur together?

In the product $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{100})$,when written in ascending powers of $x$,the highest exponent of $x$ is . . . . . . .

If $a_n = \sum_{r=0}^n \frac{1}{{}^n C_r}$,then $\sum_{r=0}^n \frac{r}{{}^n C_r} = $

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 $100k$,then $k$ is equal to $......$