$A$ question paper is divided into two parts $A$ and $B$,and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions,selecting at least two questions from each part,is:

$A$ linguistic club consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this group including the selection of a leader (from among these $4$ members) for the team. If the team has to include at most one boy,the number of ways of selecting the team is

$A$ question paper contains $4$ questions,each having $4$ alternative answers. The number of ways that a candidate can answer one or more questions is

If a seven-digit number formed with distinct digits $4, 6, 9, 5, 3, x,$ and $y$ is divisible by $3$,then the number of such ordered pairs $(x, y)$ is

In a circus,there are $10$ cages for accommodating $10$ animals. Out of these,$4$ cages are so small that $5$ out of the $10$ animals cannot enter them. In how many ways is it possible to accommodate the $10$ animals in these $10$ cages?

If $_n{P_4} = 24 \times \binom{n}{5}$,then $n = \dots$