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. Then number of different ways of displaying the three varieties of perfumes in the showcase is :

• A. $6$
• B. $25$
• C. $150$
• D. $60$

Answer 1

The correct option is C $150$

Let us assume that $V_1$, $V_2$ and $V_3$ are the three varieties of perfume.

$V_1$ $V_2$ $V_3$

Case $1$ $1$ $1$ $3$ (One triplet and two different)

Case $2$ $2$ $2$ $1$ (Two pairs and one different)

The number of different ways of displaying the three varieties of perfumes in the showcase

$=$ ${}^3{C}_1{\times }^2{C}_2\times \frac{5!}{3!}$ $+$ ${}^3{C}_1{\times }^1{C}_1\frac{5!}{2!\times 2!}$

$=\frac{5!\times 3!}{2!\times 3!}+\frac{5!\times 3!}{2!\times 2!\times 2!}$

$=60+90=150$