The total number of ways of forming a committee of $5$ members out of $7$ Indians,$6$ Americans,$5$ Russians,and $4$ Australians such that every committee contains at least one member from each country is:

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

$^nC_{r+1} + ^nC_{r-1} + 2^nC_r = \dots$

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:

If $\binom{189}{35} + \binom{189}{x} = \binom{190}{x}$,then $x = \dots$

The number of $5$ card combinations out of a deck of $52$ cards such that there is exactly one ace in each combination is: