If a person throws $3$ dice,what is the probability that the sum of the numbers obtained is exactly $15$?

$A$ bag contains $3$ red,$4$ white,and $5$ blue balls. All balls are distinct. Two balls are drawn at random. The probability that they are of different colors is

$A$ bag contains $13$ red,$14$ green,and $15$ black balls. The probability of getting exactly $2$ black balls when pulling out $4$ balls is $P_1$. Now,the number of balls of each color is doubled,and $8$ balls are pulled out. The probability of getting exactly $4$ black balls is $P_2$. Then:

From a month of $31$ days, $3$ different dates are selected at random. If the probability that these dates are in an increasing $A$.$P$. is equal to $\frac{a}{b}$, where $a, b \in N$ and $\text{gcd}(a, b) = 1$, then $a + b$ is equal to ————

If four persons are chosen at random from a group of $3$ men,$2$ women and $4$ children,then the probability that exactly two of them are children is:

$A$ bag contains $3$ red,$7$ white and $4$ black balls. If three balls are drawn from the bag,then the probability that all of them are of the same colour is