In how many ways can $8$ distinct toys be distributed among $5$ children?

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

$S_1, S_2, \ldots, S_{10}$ are the speakers in a conference. If $S_1$ addresses only after $S_2$,then the number of ways the speakers address is

The number of ways in which ten candidates $A_1, A_2, \dots, A_{10}$ can be ranked such that $A_1$ is always above $A_{10}$ is:

The number of natural numbers less than $1000$ in which no digit is repeated is

The number of ways in which $6$ boys and $4$ girls can be arranged in a row such that between any two girls there must be exactly $2$ boys is