The domain of the function $f(x) = {}^{(10-x)}C_{(x-6)}$ is...

In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is . . . . . . .

Out of $6$ boys and $4$ girls,a group of $7$ is to be formed. In how many ways can this be done if the group is to have a majority of boys?

In how many ways can $5$ letters be selected from the letters of the word $INDEPENDENT$?

If ${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$,then the minimum value of $n$ is