The sum of the digits in the unit place of all numbers formed with the help of $3, 4, 5, 6$ taken all at a time is

$5$-digit numbers are to be formed using $2, 3, 5, 7, 9$ without repeating the digits. If $p$ is the number of such numbers that exceed $20000$ and $q$ is the number of those that lie between $30000$ and $90000$,then $p : q$ is

The letters of the word $COCHIN$ are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word $COCHIN$ is

In how many ways can $6$ examination papers be arranged so that the best and the worst papers are never together?

Evaluate $\frac{n!}{(n-r)!}$,when $n=9, r=5$.

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