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

$A$ question paper has $2$ parts,part $A$ and part $B$,each containing $10$ questions. If a student has to choose $8$ questions from part $A$ and $5$ questions from part $B$,in how many ways can the student choose the questions?

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$,$Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

The number of ways in which one or more balls can be selected out of $10$ white,$9$ green,and $7$ blue balls are:

$4$ books,$1$ each in Chemistry,Physics,Biology,and Mathematics,are to be arranged on a shelf. In how many ways can this be done?

Ten students are participating in a race. In how many ways can the first $3$ prizes be won?