If $a_n = \sum_{r=0}^n \frac{1}{{}^n C_r}$,then $\sum_{r=0}^n \frac{r}{{}^n C_r} = $

Consider the following statements:
$(i)$ The number of one-one functions from set $A$ to set $B$,where $O(A) = m$ and $O(B) = n$ $(m \leq n)$,is given by ${}^n P_m$.
(ii) The number of ways in which $n$ people can be arranged at a circular table is $\frac{(n-1)!}{2}$.
(iii) The number of ways of selecting at least one thing out of the given $n$ distinct things is $2^n - 1$.
(iv) The number of ways in which $n$ distinguishable objects can be distributed into $k$ distinguishable bins is ${}^n C_{k-1}$.
Which of the following is true?

If all the numbers which are greater than $6000$ and less than $10000$ are formed with the digits $3, 5, 6, 7, 8$ without repetition of the digits,then the difference between the number of odd numbers and the number of even numbers among them is

If the number of all possible permutations of the letters of the word $MATHEMATICS$ in which the repeated letters are not together is $90(X)$,then $X=$

Four-digit numbers with all digits distinct are formed using the digits $1, 2, 3, 4, 5, 6, 7$ in all possible ways. If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than $3400$ among them,then $p: q=$

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)