$A$ box contains $5$ blue, $6$ yellow, and $4$ red balls. The number of ways of drawing $8$ balls containing at least two balls of each color is:

The number of positive integral solutions of $abc = 30$ is

Four-digit numbers with all digits distinct are formed using the digits $1, 2, 3, 4, 5, 6, 7$ in all possible ways. If $p$ is the total number of numbers thus formed and $q$ is the number of numbers greater than $3400$ among them,then $p: q=$

From $6$ different novels and $3$ different dictionaries,$4$ novels and $1$ dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$,find $x$.

The number of triplets $(x, y, z)$,where $x, y, z$ are distinct non-negative integers satisfying $x+y+z=15$,is