A box contains 3 white, 4 red and 7 blue erasers. If five erasers are taken at random then the probability that all the five are blue color is:

Answer: Option B

Total number of erasers in the box = 3 + 4 + 7 = 14
Let S be the sample space
Then, n(S) = number of ways of taking 5 out of 14
Therefore,
$$\begin{array}{rl}& \text{n}\left(\text{S}\right)={}^{14}{C}_5\\ & =\frac{14\times 13\times 12\times 11\times 10}{2\times 3\times 4\times 5}\\ & =14\times 13\times 11\\ & =2002\end{array}$$
Let E be the event of getting all the 5 blue erasers
Therefore,
$$\begin{array}{rl}& \text{n}\left(\text{E}\right)={}^7{C}_5\\ & =\frac{7\times 6\times 5\times 4\times 3}{2\times 3\times 4\times 5}\\ & =21\end{array}$$
Now, the required probability
$$\begin{array}{rl}& \frac{\text{n}\left(\text{E}\right)}{\text{n}\left(\text{S}\right)}=\frac{21}{2002}\\ & =\frac{3}{286}\end{array}$$