MCQ
Value of $\sum^{\infty}_{\text{k}=1}\sum^{\text{k}}_{\text{r}=0}\frac{1}{3^{\text{k}}}\big({^\text{k}}\text{C}_{\text{r}}\big)$ is:
- A$2$
- B$\frac{2}{3}$
- C$\frac{1}{3}$
- D$\text{None of these}$
Solution:
$\sum\frac{1}{3^{\text{k}}}{^\text{k}}\text{C}_{\text{r}}$
$=\frac{1}{3^{\text{k}}}\sum{^\text{k}}\text{C}_{\text{r}}$
$=\frac{2^{\text{k}}}{3^{\text{k}}}$
This is a G.P
Therefore, the sum of the series will be
$\text{S}=\frac{\frac{2}{3}}{1-\frac{20}{3}}=2$
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If the probability of A to fail in an examination is $\frac{1}{5}$ and that of B is$\frac{3}{10}$Then, the probability that either A or B fails is