In a library,there are $a$ books of type $A$,$2$ books of type $B$,$3$ books of type $C$,and $1$ book of type $D$. In how many ways can these books be arranged?

The number of all four-digit numbers which begin with $4$ and end with either $0$ or $5$ is:

What is the number of permutations of three letters $A, B, C$ taken all at a time?

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

Four speakers will address a meeting where speaker $Q$ will always speak before $P$. Then, the number of ways in which the order of speakers can be prepared is

If $a$ denotes the number of permutations of $x + 2$ things taken all at a time,$b$ the number of permutations of $x$ things taken $11$ at a time,and $c$ the number of permutations of $x - 11$ things taken all at a time such that $a = 182bc$,then the value of $x$ is