In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?

There are $15$ people at a party and each person shakes hands with every other person. What is the total number of handshakes?

An examiner can assign $30$ marks to $8$ questions. If he does not assign less than $2$ marks to any question,in how many ways can he assign the marks?

If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

$A$ student is allowed to select at least $(n+1)$ books but not all books from a collection of $(2n+1)$ books. If the total number of ways in which he can select these books is $255$,then the number of books in that collection is