Ten persons with badges numbered $1$ to $10$ are in a room. If three of them are asked to leave the room,then the probability that the person with the smallest badge number among the three who left is $5$,is:

Two numbers $a$ and $b$ are chosen at random from the set of the first $30$ natural numbers. The probability that $a^2 - b^2$ is divisible by $3$ is

Three mangoes and three apples are in a box. If two fruits are chosen at random,the probability that one is a mango and the other is an apple is

From a pack of $52$ playing cards,three cards are drawn simultaneously. The probability that these are one king,one queen,and one jack is:

There are $10$ bulbs in a room,out of which $4$ are defective. If $3$ bulbs are chosen at random,what is the probability that the room will be illuminated? (The room is illuminated if at least one bulb is good.)

Two different families $A$ and $B$ are blessed with an equal number of children. There are $3$ tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of family $B$ is $\frac{1}{12}$,then the number of children in each family is?