In how many ways can $n$ prizes be distributed among $n$ boys such that no boy gets all the prizes?

If $4$-digit numbers greater than $5,000$ are randomly formed from the digits $0, 1, 3, 5,$ and $7$,what is the probability of forming a number divisible by $5$ when the repetition of digits is not allowed?

Consider the following statements:
$i.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins $(k \leq n)$ such that no bin is empty is ${}^{n-1}C_{k-1}$.
$ii.$ The number of ways of writing a positive integer $n$ as a sum of $k$ positive integers is ${}^{n-1}C_{k-1}$.
$iii.$ The number of ways of placing $n$ distinct objects in $k$ distinct bins such that at least one bin is non-empty is ${}^{n-1}C_{k-1}$.
$iv.$ ${}^nC_k - {}^{n-1}C_k = {}^{n-1}C_{k-1}$.

If $^nC_r = ^nC_{r-1}$ and $^nP_r = ^nP_{r+1}$,then the value of $n$ is

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition such that the number is divisible by $3$?

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $x$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,