In an apartment,there are $30$ kids. If each kid plays table tennis with every other kid exactly once,then the total number of matches played by them is:

If at the end of a certain meeting,everyone had shaken hands with everyone else,and it was found that $45$ handshakes were exchanged,then the number of members present at the meeting is:

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that the team includes exactly $3$ bowlers and $1$ wicket-keeper?

If ${ }^9 C_3+{ }^9 C_5={ }^{10} C_r$ for some $r \in N$,then $r=$

If $^nC_4, ^nC_5,$ and $^nC_6$ are in $A.P.,$ then $n$ can be

The number of all possible ways of distributing $8$ identical apples among $3$ persons is