${ }^{34} C_5+\sum_{r=0}^4{ }^{(38-r)} C_4=$

How many numbers can be formed from the digits $1, 2, 3, 4$ when repetition is not allowed?

The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

Consider the following two statements:
$I.$ If $n$ is a composite number,then $n$ divides $(n-1)!$.
$II.$ There are infinitely many natural numbers $n$ such that $n^3+2n^2+n$ divides $n!$.

Numbers between $1$ and $10,000$ are formed using the digits $2$ and $3$ exactly once and the digit $4$ twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of $4324$ and $324$ respectively,then $x-y=$

$A$ five-digit number divisible by $3$ has to be formed using the numerals $0, 1, 2, 3, 4,$ and $5$ without repetition. The total number of ways in which this can be done is: