There are $5$ apples,$4$ mangoes,$3$ oranges,and $1$ each of $2$ other varieties of fruits. The number of ways of selecting at least one fruit of each kind is

Statement-$1:$ The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3$.
Statement-$2:$ The number of ways of choosing any $3$ places from $9$ different places is $^9C_3$.

If $x, y$ and $r$ are positive integers,then $^x{C_r} + ^x{C_{r-1}} ^y{C_1} + ^x{C_{r-2}} ^y{C_2} + \dots + ^y{C_r} = $

The number of ways in which $17$ apples can be distributed among $4$ guests such that each guest gets at least $3$ apples is

In a conference of $8$ persons,if each person shakes hands with every other person exactly once,then the total number of handshakes is:

If ${}^{10}P_r = 604800$ and ${}^{10}C_r = 120$,then $r =$