If $4 \times ^{15}P_r = 3 \times ^{16}P_{r-1}$,then $r = \dots$

$10$ different letters of the English alphabet are given. Out of these letters,words of $5$ letters are formed. How many words can be formed if at least one letter is repeated?

There are $9$ balls to be placed in $9$ boxes. If $5$ of the balls cannot fit into $3$ specific small boxes,find the number of ways to arrange exactly one ball in each of the boxes.

Total number of four-digit odd numbers that can be formed using $0, 1, 2, 3, 5, 7$ (repetition is allowed) is:

If all the words formed by using the letters of the word $CRICKET$ are arranged in a dictionary in alphabetical order,how many words appear before the word $CRICKET$?

Each of the $10$ letters $A, H, I, M, O, T, U, V, W$ and $X$ appears the same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many $3$-letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter?