The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

The number of ways of selecting $8$ letters from $24$ letters,of which $8$ are $a$,$8$ are $b$,and the remaining $8$ are distinct,is given by:

Consider the set $A = \{1, 2, 3, \ldots, 30\}$. The number of ways in which one can choose three distinct numbers from $A$ such that the product of the chosen numbers is divisible by $9$ is:

For a natural number $n$,$(n!)^2 > n^n$ is true if:

The number of numbers greater than $5000$ and less than $9000$ that are divisible by $3$, which can be formed using the digits $0, 1, 2, 5, 9$ with repetition allowed, is . . . . . . .

If $8 \cdot {}^{7}P_{r} = 7 \cdot {}^{8}P_{r-1}$,what is the value of $r$?