The number of ways a committee of $8$ members can be formed from a group of $10$ men and $8$ women such that the committee contains at most $5$ men and at least $5$ women 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 of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2$ and $z \geq 5$ is:

$^{2n}C_4 : ^nC_3 = 99 : 4 \Rightarrow n = $

$A$ bag contains $3$ coins of one rupee,$4$ coins of fifty paise,and $5$ coins of ten paise. If at least one coin is selected from the bag,what is the total number of ways to make the selection?