Out of $8$ students in a classroom,$4$ of them are chosen and they are arranged around a table. If the remaining $4$ are arranged in a row,then the total number of arrangements that can be made with those $8$ students is

The total number of numbers lying between $100$ and $1000$ that can be formed with the digits $1, 2, 3, 4, 5$,if the repetition of digits is not allowed and the numbers are divisible by either $3$ or $5$,is:

The number of ordered pairs $(x, y)$ of integers such that their product $xy$ is a positive integer less than $100$ is:

Consider $4$ boxes,where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

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?

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 $p: q=$