The number of ways to give away $25$ apples to $4$ boys, each boy receiving at least $4$ apples, are

The number of ways in which we can select three numbers from $1$ to $30$ so as to exclude every selection of all even numbers is

If $\sum\limits_{i = 0}^4 {^{4 + 1}} {C_i} + \sum\limits_{j = 6}^9 {^{3 + j}} {C_j} = {\,^x}{C_y}$ ($x$ is prime number), then which one of the following is incorrect 

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$, if all digits are to appear in the same number at least once, is

If $^{15}{C_{3r}}{ = ^{15}}{C_{r + 3}}$, then the value of $r$ is