Prove that the minimum value of $${{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},$$
$$a,b > c,x > - c$$ is $${\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}$$.

Evaluate $$\int {{{{x^2}dx} \over {{{\left( {a + bx} \right)}^2}}}} $$

Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:

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.