In how many different ways can the letters of the word '$THERAPY$' be arranged so that the vowels never come together?

How many odd numbers of four digits can be formed with the digits $0, 1, 2, 3, 4, 7$ and $8$?

How many $2$-digit numbers are there with distinct digits?

There are $16$ points in a plane,no three of which are in a straight line except $8$ which are all in a straight line. The number of triangles that can be formed by joining them equals

If $x, y$ and $r$ are positive integers, then $^x{C_r} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^y{C_r} = $

Consider three boxes,each containing $10$ balls labelled $1, 2, dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is: