Out of $6$ books,in how many ways can a set of one or more books be chosen?

$A$ person $X$ has four notes of $1, 2, 5,$ and $10$ denominations. The number of different sums of money she can form from them is

There are $10$ different letters of the English alphabet. If words of $5$ letters are formed using these letters,how many words can be formed such that at least one letter is repeated?

$A$ polygon has $35$ diagonals; then the number of its sides is:

The value of $r$ for which $^{15}C_{r + 3} = {}^{15}C_{2r - 6}$ 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?