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:

If $7$ identical white balls and $3$ identical black balls are arranged in a row at random,what is the probability that no two black balls are together?

$A$ poker hand consists of $5$ cards drawn at random from a well-shuffled pack of $52$ cards. The probability that a poker hand consists of a pair and a triple of equal face values (for example, $2$ sevens and $3$ kings or $2$ aces and $3$ queens, etc.) is:

If two numbers $a$ and $b$ are chosen from the first $30$ natural numbers,what is the probability that $a^2 - b^2$ is divisible by $3$?

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

$A$ bag contains $3$ red,$4$ white,and $5$ black balls. Three balls are drawn at random. The probability that they are of different colours is