How many groups can be selected for playing football out of $5$ ladies and $4$ gentlemen such that there should be one lady and one gentleman on each side?

There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

The letters of the word $MODESTY$ are written in all possible orders and these words are written out as in a dictionary,then the rank of the word $MODESTY$ is

The straight lines $l_1, l_2, l_3$ are parallel and lie in the same plane. $A$ total number of $m$ points are taken on $l_1$,$n$ points on $l_2$,and $k$ points on $l_3$. The maximum number of triangles formed with vertices at these points is:

If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio $2:1$,find the value of $n$.

There are $3$ candidates for a post and one is to be selected by the votes of $7$ men. The number of ways in which votes can be given is