Question

Three mangoes and three apples in a box. If two fruits are chosen at random, the probability that one is a mango and the other is an apple is

• A. $\frac{2}{3}$
• B. $\frac{3}{5}$
• C. $\frac{1}{3}$
• D. None of these

Answer 1

The correct option is B

$\frac{3}{5}$

Explanation for the correct answer :

Given: Three mangoes and three apples.

Total fruits are $6$

Sample space for choosing two fruits from $6$ is ${}^6{C}_2$

The number of ways to select one mango and one apple is ${}^3{C}_1\times {}^3{C}_1$

Therefore, probability of choosing two fruits such that one is mango and one is apple is $\frac{{}^3{\mathrm{C}}_1\times {}^3{\mathrm{C}}_1}{{}^6{\mathrm{C}}_2}$

$$\begin{gathered} =\frac{3\times 3}{{\displaystyle \frac{6!}{2\left(6-2\right)!}}}\left[{}^{\mathrm{n}}{\mathrm{C}}_1=\mathrm{n}\mathrm{and}{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}=\frac{\mathrm{n}!}{\mathrm{r}!\left(\mathrm{n}-\mathrm{r}\right)!}\right] \\ =\frac{9}{{\displaystyle \frac{6\times 5\times 4!}{2\times 4!}}} \\ =\frac{9}{15} \\ =\frac{3}{5} \end{gathered}$$

Hence option B is the correct answer.