Consider the matrix $A$ below:
$$ A=\left[\begin{array}{llll} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{array}\right] $$
For which of the following combinations of $\alpha, \beta$ and $\gamma$, is the rank of $A$ at least three?
(i) $\alpha=0$ and $\beta=\gamma \neq 0$
(ii) $\alpha=\beta=\gamma=0$
(iii) $\beta=\gamma=0$ and $\alpha \neq 0$
(iv) $\alpha=\beta=\gamma \neq 0$
Consider the following series:
(i) $\sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n}}$
(ii) $ \sum\limits_{n=1}^{\infty} \frac{1}{n(n+1)}$
(iii) $\sum\limits_{n=1}^{\infty} \frac{1}{n!}$
A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.
What is the probability that the two balls drawn have different colours?