On the occasion of the Deepawali festival,each student of a class sends greeting cards to all the others. If there are $20$ students in the class,then the total number of greeting cards exchanged by the students is:

In how many different ways can the letters of the word '$BANKING$' be arranged so that the vowels always come together?

If the ratio $C(2n, 3) : C(n, 3)$ is equal to $11 : 1$,find $n$.

In an examination,Yamini has to select $4$ questions from each part. There are $6, 7,$ and $8$ questions in Part-$I,$ Part-$II,$ and Part-$III,$ respectively. What is the number of possible combinations in which she can choose the questions?

In a plane there are $37$ straight lines of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. Then the number of intersection points the lines have is equal to

It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?