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:

The probability of forming a $12$-person committee from $4$ engineers,$2$ doctors,and $10$ professors containing at least $3$ engineers and at least $1$ doctor is:

Three squares of a chessboard are selected at random. The probability of selecting two squares of one colour and the other of a different colour is equal to

$A$ bag contains $6$ white and $4$ black balls. Two balls are drawn at random. The probability that they are of the same colour is

$A$ drawer contains $5$ brown socks and $4$ blue socks well mixed. $A$ man reaches the drawer and pulls out $2$ socks at random. What is the probability that they match?

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 ————