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

There are $4$ oranges,$5$ apples,and $7$ mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

$A$ scientific committee is to be formed from $6$ Indians and $8$ foreigners,which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways,the committee can be formed,is

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

$A$ man has $10$ friends. In how many ways can he invite one or more of them to a party?

The number of ways in which $15$ identical gold coins can be distributed among $3$ persons such that each one gets at least $3$ gold coins is