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

Five students are selected from $n$ students such that the ratio of the number of ways in which $2$ particular students are selected to the number of ways $2$ particular students are not selected is $2:3$. Then the value of $n$ is

If a committee of $10$ members is to be formed from $8$ men and $6$ women,then the number of different possible committees in which the men are in majority is

$A$ father with $8$ children takes $3$ children to the zoo at a time. If he can go more than once without taking the same $3$ children together,in how many ways can the father go to the zoo?

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. The number of choices available to him is

In an election,a voter can vote for any number of candidates but not more than the number of candidates to be elected. There are $10$ candidates and $4$ are to be elected. If a voter must vote for at least one candidate,in how many ways can they vote?