If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

How many words,each of $3$ vowels and $2$ consonants,can be formed from the letters of the word '$INVOLUTE$'?

There are $n$ straight lines in a plane,no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

There are $10$ points in a row. In how many ways can $4$ points be selected such that no two of them are consecutive?

If the four-letter words (need not be meaningful) are to be formed using the letters from the word "$MEDITERRANEAN$" such that the first letter is $R$ and the fourth letter is $E,$ then the total number of all such words is

If $33!$ is divisible by $2^n$,where $n \in N$,then the sum of all possible values of $n$ is: