It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

Let $A_1, A_2, ......, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, ......, 11$. One hundred gold coins were won by the team. These coins are to be distributed among the players such that each player $A_i$ gets at least $i$ coins,plus an additional coin (i.e.,at least $i+1$ coins). Additionally,the captain (assume $A_1$) and vice-captain (assume $A_2$) must receive at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. In how many different ways can these coins be distributed?

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

The number of ways to distribute $25$ apples among $4$ boys,such that each boy receives at least $4$ apples,is:

The number of ways that $8$ beads of different colours can be strung as a necklace is:

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?