$^nC_r + ^{n-1}C_r + ...... + ^rC_r$ =

There are $9$ chairs in a room on which $6$ persons are to be seated. Out of these $6$ persons,one is a guest who must sit on one specific chair. In how many ways can they be seated?

If $^nC_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression $^nC_{r+1} + ^nC_{r-1} + 2 \times ^nC_r$ equals

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:

The solution set of $^{10}C_{x-1} > 2 \cdot ^{10}C_x$ is

How many $3$-letter words can be formed from the letters of the word '$EXPERTISE$',if repetition of letters is not allowed?