For a set of $5$ true or false questions,no student has written all the correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

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

The number of ways in which a committee of $7$ members can be formed from $6$ teachers,$5$ fathers,and $4$ students in such a way that at least one from each group is included and teachers form the majority among them,is

Out of $7$ consonants and $4$ vowels,words are formed each having $3$ consonants and $2$ vowels. The number of such words that can be formed is

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

$A$ man has $7$ relatives,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,is: