If $C(n, 7) = C(n, 5)$,find $n$.

How many $3$-digit numbers can be formed by using the digits $0, 2, 3, 6, 8$ when the digits may be repeated any number of times?

The number of words formed from the letters of the word $RAJASTHAN$ by taking all the letters at a time in which vowels are alternate 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?

The total number of possible ways in which a person can go from point $A(0,0)$ to point $B(3,3)$ if they can travel only to the right or upward along the grid lines is equal to:

$6$ points in a plane are joined in all possible ways by indefinite straight lines. If no two lines are coincident or parallel,and no three lines pass through the same point (with the exception of the original $6$ points),the number of distinct points of intersection is equal to