In how many ways can $11$ identical pencils be distributed among $6$ children such that each child receives at least one pencil?

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

$A$ student is allowed to select at least $(n+1)$ books but not all books from a collection of $(2n+1)$ books. If the total number of ways in which he can select these books is $255$,then the number of books in that collection is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are two red cards and two black cards?

The total number of all those $3$-digit numbers in which the sum of all the digits in each of them is $10$,is

In an examination of $9$ papers,a candidate is successful only if they pass in more papers than they fail. In how many ways can the candidate be unsuccessful?