Cards are drawn one by one at random from a well - shuffled full pack of $$52$$ playing cards until $$2$$ aces are obtained for the first time. If $$N$$ is the number of cards required to be drawn, then show that $${P_r}\left\{ {N = n} \right\} = {{\left( {n - 1} \right)\left( {52 - n} \right)\left( {51 - n} \right)} \over {50 \times 49 \times 17 \times 13}}$$ where $$2 \le n \le 50$$

$$A$$ and $$B$$ are two candidates seeking admission in $$IIT.$$ The probability that $$A$$ is selected is $$0.5$$ and the probability that both $$A$$ and $$B$$ are selected is atmost $$0.3$$. Is it possible that the probability of $$B$$ getting selected is $$0.9$$ ?

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $$0.4, 0.3, 0.2$$ and $$0.1$$ respectively. What is the probability that the gun hits the plane?

Six boys and six girls sit in a row randomly. Find the probability that
(i) the six girls sit together
(ii) the boys and girls sit alternately.