The number of ways $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together,is $......$

In a group of $3$ girls and $4$ boys,there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together,all the boys stand together,but $B_1$ and $B_2$ are not adjacent to each other,is:

The remainder obtained when $1! + 2! + 3! + \dots + 11!$ is divided by $12$ is

The number of ways of selecting two numbers $a$ and $b$,where $a \in \{2, 4, 6, \ldots, 100\}$ and $b \in \{1, 3, 5, \ldots, 99\}$,such that the remainder is $2$ when $a+b$ is divided by $23$ is:

If $x$ and $y$ are digits such that $17! = 355687428096000$, then $x+y$ equals

In the product $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{100})$,when written in ascending powers of $x$,the highest exponent of $x$ is . . . . . . .