How many words can be formed using the letters of the word $MOBILE$ such that the consonants always occupy odd positions?

If the different permutations of all the letters of the word $EXAMINATION$ are listed as in a dictionary,how many words are there in this list before the first word starting with $E$?

Three persons enter a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. If the lift does not stop at the $1^{\text{st}}$,$2^{\text{nd}}$,and $3^{\text{rd}}$ floors,the number of ways in which the three persons can exit the lift at three different floors is equal to . . . . . . .

If $a$ denotes the number of permutations of $(x + 2)$ items taken all at a time,$b$ denotes the number of permutations of $x$ items taken $11$ at a time,and $c$ denotes the number of permutations of $(x - 11)$ items taken all at a time. If $a = 182bc$,find the value of $x$.

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 $70513$?

All the letters of the word $GTWENTY$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $GTWENTY$ is: