The value of $r$ for which $^{15}C_{r+3} = ^{15}C_{2r-6}$ is

There are $15$ persons in a party and each person shakes hands with every other person exactly once. The total number of handshakes is:

If the number of subsets with $8$ elements from the set $A=\{a_1, a_2, a_3, \ldots, a_n\}$,$n \geq 8$,is five times the number of such subsets containing $a_4$,then $n=$

The number of ways of choosing a committee from $4$ men and $6$ women so that the committee includes at least $2$ men and exactly twice as many women as men 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:

Out of $10$ white,$9$ black and $7$ red balls,the number of ways in which selection of one or more balls can be made,is