The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$,if all digits are to appear in the same number at least once,is

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $

The number of all possible positive integral solutions of the equation $xyz=30$ is

The number of different signals which can be given from $7$ different coloured sheets,taking one or more at a time is

The number of non-negative integer solutions of the equations $6x + 4y + z = 200$ and $x + y + z = 100$ is

$A$ man $P$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Q$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $P$ and $Q$ have no common friends. Then the total number of ways in which $P$ and $Q$ together can throw a party inviting $3$ ladies and $3$ men,so that $3$ friends of each of $P$ and $Q$ are in this party,is . . . . . . .