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:

Two integers are drawn at random from the set $\{5, 6, \ldots, 35\}$. What is the probability that their difference is odd?

There are $10$ engineering colleges and five students $A, B, C, D, E$. Each of these students received an offer from all of these $10$ engineering colleges. They randomly choose a college independently of each other. The probability that all get admission in different colleges can be expressed as $\frac{a}{b}$,where $a$ and $b$ are co-prime numbers. Then the value of $a + b$ is:

Two numbers $x$ and $y$ are chosen at random (without replacement) from the set $\{1, 2, 3, \dots, 1000\}$. The probability that $|x^4 - y^4|$ is divisible by $5$ is -

If two cards are drawn at random simultaneously from a well-shuffled pack of $52$ playing cards, then the probability of getting a card having a composite number and a card having a number which is a multiple of $3$ is

If $6$ boys and $3$ girls are to be seated on $9$ chairs for a photograph,then the probability that the end seats are occupied by the girls and no two girls are side by side is