In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?

The value of $^{15}C_{3} + ^{15}C_{13}$ is

Let $A_1, A_2, \dots, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, \dots, 11$. One hundred gold coins were won by the team in the final match. These coins are to be distributed among the players such that each player $A_i$ gets at least $i+1$ coins,except for the captain and vice-captain who get at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. If the captain is $A_1$ and the vice-captain is $A_2$,in how many different ways can these coins be distributed?

If ${ }^{15} C_4+{ }^{15} C_5+{ }^{16} C_6+{ }^{17} C_7+{ }^{18} C_8={ }^{19} C_{r}$,then the value of $r$ is equal to

The number of elements in the set $\{x \in N : ^{20-2x}C_{x-3} \in N\}$ is

The number of ways in which $6$ distinct things can be distributed into $2$ boxes so that no box is empty is