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

The number of integers between $10$ and $10,000$ such that in every integer every digit is greater than its immediate preceding digit,is

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

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

$A$ question paper consists of two sections having $3$ and $4$ questions respectively. The following note is given on the paper: "It is not necessary to attempt all the questions. One question from each section is compulsory". In how many ways can a candidate select the questions?

The total number of $3$-digit numbers,whose sum of digits is $10$,is