The number of integers between $10$ and $10,000$ such that in every integer every digit is greater than its immediate preceding digit,is

From a group of $10$ men and $8$ women,the number of ways of forming a committee of $8$ members with not more than $5$ men and not less than $5$ women is

There are $7$ identical white balls and $3$ identical black balls. The number of distinguishable arrangements in a row of all the balls,so that no two black balls are adjacent is

If $\frac{1}{{ }^{5}C_{r}} + \frac{1}{{ }^{6}C_{r}} = \frac{1}{{ }^{4}C_{r}},$ then the value of $r$ is

$A$ scientific committee is to be formed from $6$ Indians and $8$ foreigners,which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways,the committee can be formed,is

The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is