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

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)

In how many ways can a team of $3$ boys and $3$ girls be selected from $5$ boys and $4$ girls?

The number of ways of distributing $500$ dissimilar boxes equally among $50$ persons is:

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men. The number of committees in which the women are in majority is

The value of $r$ for which $^{20}C_r ^{20}C_0 + ^{20}C_{r-1} ^{20}C_1 + ^{20}C_{r-2} ^{20}C_2 + ... + ^{20}C_0 ^{20}C_r$ is maximum is