In how many ways can $5$ speakers $S_1, S_2, S_3, S_4$,and $S_5$ give speeches one after the other if $S_3$ wants to speak after $S_1$ and $S_2$?

$A$ student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first $5$ questions. The number of choices available to him is

The set $S = \{1, 2, 3, \ldots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size. Thus $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is

$P_1$ and $P_2$ are two distinct and intersecting planes. Three non-collinear points lie on $P_1$ and another three non-collinear points lie on $P_2$ (none being on the line of intersection of the planes). Then,the maximum number of tetrahedrons formed using these six points is:

In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

How many triangles can be drawn by using $9$ non-collinear points?