There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included,and $G$ and $H$ must not be included in the team of $5$?

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

If $\binom{a^2+a}{3} = \binom{a^2+a}{9}$,then $a = \dots$

If a question paper consists of $11$ questions divided into two sections,$I$ and $II$. Section $I$ consists of $6$ questions and section $II$ consists of $5$ questions,then the number of different ways a student can select $6$ questions,taking at least $2$ questions from each section,is

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

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