$A$ lock is made with $3$ rings,each marked with $10$ different letters. How many unsuccessful attempts can be made to open the lock?

There are $10$ distinct letters of the English alphabet. Words of $5$ letters are formed using these letters. How many such words can be formed if at least one letter is repeated?

How many $5$ digit telephone numbers can be constructed using the digits $0$ to $9$,if each number starts with $67$ and no digit appears more than once?

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 5, 7$ without repetition? If all such numbers are arranged in ascending order,what is the rank of the number $70513$?

$5$ men and $4$ women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $\alpha$ and the number of arrangements in which those two are not together is $\beta$,then $\alpha: \beta=$

Evaluate $\frac{n!}{(n-r)!}$ when $n=6$ and $r=2$.