$A$ man has $7$ friends. In how many ways can he invite one or more of them for a tea party?

There are $12$ volleyball players in all in a college,out of which a team of $9$ players is to be formed. If the captain always remains the same,then in how many ways can the team be formed?

The number of $5$ digit telephone numbers having at least one of their digits repeated is

In an examination hall,there are $4$ rows of chairs. Each row has $8$ chairs,$1$ behind the other. There are $2$ classes sitting for the examination with $16$ students in each class. It is desired that in each row,all students belong to the same class and that no $2$ adjacent rows are allotted to the same class. In how many ways can these $32$ students be seated?

How many words can be formed by using all the letters of the word '$TRIVANDRUM$' using each letter exactly once (in $!$)?

There are $6$ multiple choice questions on an examination. How many sequences of answers are possible,if the first three questions have $4$ choices each and the next $3$ have $5$ each?