Twenty tickets are marked the numbers $1, 2, ..... 20.$ If three tickets be drawn at random, then what is the probability that those marked $7$ and $11$ are among them

Twenty persons arrive in a town having $3$ hotels $x, y$ and $z$. If each person randomly chooses one of these hotels, then what is the probability that atleast $2$ of them goes in hotel $x$, atleast $1$ in hotel $y$ and atleast $1$ in hotel $z$ ? (each hotel has capacity for more than $20$ guests)

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, then the probability that $\omega^{I_1}+\omega^{\mathrm{I}_2}+\omega^{\mathrm{I}_3}=0$ is

Find the probability that when a hand of $7$ cards is drawn from a well shuffled deck of $52$ cards, it contains $3$ Kings.

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p : q = m$ $: n$, where $m$ and $n$ are coprime, then $m + n$ is equal to $..........$.

A box contains $10$ red marbles, $20$ blue marbles and $30$ green marbles. $5$ marbles are drawn from the box, what is the probability that all will be blue?