Fifteen persons,including $A$ and $B$,sit down at random at a round table. The probability that there are $4$ persons between $A$ and $B$ is

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:

$A$ bag contains $6$ blue and $6$ green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2,$ and $r_3$ are the numbers obtained on the die,what is the probability that $\omega^{r_1} + \omega^{r_2} + \omega^{r_3} = 0$?

Three numbers are chosen at random from $1$ to $20$. The probability that the sum of the three numbers is divisible by $3$ is:

If a coin is tossed seven times,then the probability of getting exactly three heads such that no two heads occur consecutively is