The number of seven-digit positive integers formed using the digits $1, 2, 3,$ and $4$ only,such that the sum of the digits is equal to $12$,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

$A$ 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 . . . . . . .

From $6$ different novels and $3$ different dictionaries,$4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

The number of ways $5$ boys and $4$ girls can sit in a row so that either all the boys sit together or no two boys sit together,is $......$

The number of different words that can be formed from the letters of the word "$INTERMEDIATE$" such that two vowels never come together,is