A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If two marbles are picked at random, what is the probability that they are either blue or yellow?

Answer: Option C

Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles.
Probability that both marbles are blue
$$\begin{array}{rl}& =\frac{{}^3{C}_2}{{}^{15}{C}_2}\\ & =\frac{3\times 2}{15\times 14}\\ & =\frac{1}{35}\end{array}$$
Probability that both are yellow
$$\begin{array}{rl}& =\frac{{}^2{C}_2}{{}^{15}{C}_2}\\ & =\frac{2\times 1}{15\times 14}\\ & =\frac{1}{105}\end{array}$$
Probability that one blue and other is yellow
$$\begin{array}{rl}& =\frac{{}^3{C}_1\times {}^2{C}_1}{{}^{15}{C}_2}\\ & =\frac{2\times 3\times 2}{15\times 14}\\ & =\frac{2}{35}\end{array}$$
∴ Required probability
$$\begin{array}{rl}& \frac{1}{35}+\frac{1}{105}+\frac{2}{35}\\ & =\frac{3+1+6}{105}\\ & =\frac{10}{105}\\ & =\frac{2}{21}\end{array}$$