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

The number of integral solutions to the equation $x+y+z=21$,where $x \geq 1, y \geq 3, z \geq 4$,is equal to $..........$.

$^{14}C_4 + \sum_{j=1}^{4} {^{18-j}C_3}$ is equal to

Let $S = \{1, 2, 3, \ldots, 9\}$. For $k = 1, 2, \ldots, 5$,let $N_k$ be the number of subsets of $S$,each containing five elements out of which exactly $k$ are odd. Then $N_1 + N_2 + N_3 + N_4 + N_5 =$

$n$ balls each of weight $w$. When weighed in pairs,the sum of the weights of all possible pairs is $120$. When they are weighed in triplets,the sum of the weights of all possible triplets is $480$. Then $n$ is:

If $\binom{a^2+a}{3} = \binom{a^2+a}{9}$,then $a = \dots$