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

In how many ways can a committee of $6$ members be formed out of $10$ members,such that it always includes a specified member?

$A$ group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least one boy and one girl?

If $\binom{n+1}{3} = 2 \times \binom{n}{2}$,then $n = \dots$

There are $5$ types of ice creams available in a shop. $A$ child buys $6$ ice creams.
Statement-$1$: The child can buy $6$ ice creams in $^{10}C_5$ ways.
Statement-$2$: The number of ways the child can buy $6$ ice creams is equal to the number of ways to arrange $6$ '$A$'s and $4$ '$B$'s in a row.

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?