At an election,a voter may vote for any number of candidates,not greater than the number to be elected. There are $10$ candidates and $4$ are to be selected. If a voter votes for at least one candidate,then the number of ways in which he can vote is

In how many ways can $12$ gentlemen sit around a round table so that three specified gentlemen are always together?

The number of ways in which $6$ distinct rings can be worn on the $4$ fingers of one hand is:

If $P(56, r+6) : P(54, r+3) = 30800 : 1$,find $r$.

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

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?