Question

A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in this stock. There are $5$ places in a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the showcase is

• A. $6$
• B. $50$
• C. $150$
• D. None of these

Answer 1

The correct option is C

$150$

Explanation for the correct option:

Step 1. Given data

A shopkeeper sells three varieties of perfumes and there are $5$ places in a row in his showcase

Step 2. Finding the number of ways of displaying

Therefore, the possible number of ways that he can display three varieties of perfumes is

$X=3^5$

The number of ways of occupying a single variety is

$Y=3\times 1$

As there will be two ways for all places to be occupied by a single variety each and the number of ways of occupying by two different varieties is

$Z={}^{3}C_2\left(2^5-2\right)$

Step 3. Total number of ways of displaying all three varieties

The total number of ways of displaying all three varieties are

$$\begin{gathered} =X-Y-Z \\ =3^5-3-{}^{3}C_2\left(2^5-2\right) \\ =243-3-90 \\ =150 \end{gathered}$$

Therefore, the correct option is (C).