Question
A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his 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 :
150
Solution :
\[\begin{gathered}
\begin{array}{*{20}{c}}
{Possibilities}&{Selections}&{Arrangements} \\
{{\text{One triplet, two different}}}&{^3{C_1} \times {\,^2}{C_2}}&{^3{C_1} \times {\,^2}{C_2} \times \frac{{5!}}{{3!}} = 60} \\
{{\text{Two pairs, one different}}}&{^3{C_2} \times {\,^1}{C_1}}&{\underline {^3{C_2} \times {\,^1}{C_1} \times \frac{{5!}}{{2!\,2!}} = 90} }
\end{array} \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore \,\,{\text{the required number of ways}} = 150. \hfill \\
\end{gathered} \]