The number of all $5$-digit numbers with distinct digits is:

In a train,five seats are vacant. In how many ways can three passengers sit?

$n$-digit numbers are formed using only three digits $2, 5,$ and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed 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

In how many of the distinct permutations of the letters in $MISSISSIPPI$ do the four $I$'s not come together?

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?