If ${ }^{18} C_{r}={ }^{18} C_{r+2}$,find ${ }^{r} C_{5}$.

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is

Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is

The number of four-digit natural numbers which contain exactly two distinct digits is:

In how many ways can a cricket team of $11$ players be selected out of $16$ players,if $2$ particular players are always to be included?

Out of $5$ apples,$10$ mangoes,and $15$ oranges,$15$ fruits are to be distributed between two persons. Find the total number of ways of distribution.