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 six guests. The number of ways in which they may be selected from among ten friends,if two of the friends will not attend the party together is

$A$ number lock on a suitcase has $3$ wheels,each labelled with $10$ digits from $0$ to $9$. If opening the lock requires a particular sequence of $3$ digits with no repeats,how many such sequences are possible?

In how many ways can a team of $11$ members be formed from a group of $15$ persons?

There are $12$ volleyball players in all in a college,out of which a team of $9$ players is to be formed. If the captain always remains the same,then in how many ways can the team be formed?

The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to