The total number of $4$-digit numbers which can be formed using the digits $1, 2, 3, 4$ without repetition such that the digit $n+1$ never immediately follows the digit $n$ is:

$A$ three-digit number $n$ is such that the last two digits are equal and differ from the first digit. The number of such $n$'s is:

$6$ boys and $5$ girls sit in a line such that $(I)$ no two girls sit together $(II)$ all the girls sit together. If $p$ is the number of arrangements in case $(I)$ and $q$ is the number of arrangements in case $(II)$,then $p/q =$

How many numbers greater than a million can be formed with the digits $2, 3, 0, 3, 4, 2, 3$?

Find $r$ if $^{5}P_{r} = 2 \times ^{6}P_{r-1}$.

Assuming that no two consecutive digits are same,the number of $n$-digit numbers is: