The number of ways in which $6$ distinct things can be distributed into $2$ boxes so that no box is empty is

There are $10$ intermediate stations on a railway line between two particular stations. The number of ways that a train can be made to stop at $3$ of these intermediate stations so that no two of these halting stations are consecutive is:

The value of ${ }^{16} C_{9}+{ }^{16} C_{10}-{ }^{16} C_{6}-{ }^{16} C_{7}$ is

The number of integers between $100$ and $1000$ such that the sum of their digits is equal to $14$ is:

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} = $

If $\binom{15}{3r} = \binom{15}{r+3}$,then $r = \dots$