Match the following:
If $ \sum_{r=0}^{10} \left( 10^{r+1} - 1 \right)$ $\,$\(\binom{10}{r} = \alpha^{11} - 1 \), then $ \alpha $ is equal to :
Given three identical bags each containing 10 balls, whose colours are as follows:
Bag I | 3 Red | 2 Blue | 5 Green |
Bag II | 4 Red | 3 Blue | 3 Green |
Bag III | 5 Red | 1 Blue | 4 Green |
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is: