$4$ books,$1$ each in Chemistry,Physics,Biology,and Mathematics,are to be arranged on a shelf. In how many ways can this be done?

The number of ways of selecting $15$ teams from $15$ men and $15$ women,such that each team consists of a man and a woman,is

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is

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

The total number of seven-digit numbers the sum of whose digits is even is

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves exceeded the number of games that the men played with the women by $66$. The total number of participants is: