If $x$ and $y$ are digits such that $17! = 355687428096000$, then $x+y$ equals

If $x={ }^{16} C_5+{ }^{12} C_4, y=\sum_{r=1}^3{ }^{(20-r)} C_4, z=\sum_{k=1}^4{ }^{(16-k)} C_3$,then $x+y+z=$

Let $S = \{0, 1, 2, 3, \ldots, 100\}$. The number of ways of selecting $x, y \in S$ such that $x \neq y$ and $x + y = 100$ is

Total number of $3$ letter words that can be formed from the letters of the word $'SAHARANPUR'$ is equal to

If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

The students $S_{1}, S_{2}, \ldots, S_{10}$ are to be divided into $3$ groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is ........ .