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

$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.

Consider a group of $5$ boys and $7$ girls. The number of different teams,consisting of $2$ boys and $3$ girls that can be formed from this group if there are two specific girls $A$ and $B$,who refuse to be the members of the same team,is

The least value of $n$ for which ${ }^{(n-1)} C_2 + { }^{(n-1)} C_3 > { }^n C_2$ is

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is . . . . . . .