Problem Solving

Source: GMAT Prep

To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from each other. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?

A. 6
B. 8
C. 10
D. 15
E. 30

The OA is A.

BTGmoderatorLU wrote:Source: GMAT Prep

To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from each other. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?

A. 6
B. 8
C. 10
D. 15
E. 30

\[? = T\,\,\,\left( {\# \,\,{\text{tables}}} \right)\]
\[10 \cdot C\left( {T,2} \right) = C\left( {5,2} \right) \cdot C\left( {T,2} \right) = 150\,\,\,\,\,\, \Rightarrow \,\,\,\,\,C\left( {T,2} \right) = 15\]
\[C\left( {T,2} \right) = 15\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\left\{ \begin{gathered}
T\left( {T - 1} \right) = 30 \hfill \\
6 \cdot 5 = 30\,\,\,\,\left( {**} \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = T = 6\]
\[\left( * \right)\,\,\,C\left( {T,2} \right) = \underleftrightarrow {\frac{{T\left( {T - 1} \right)\left( {T - 2} \right)!}}{{2!\,\,\left( {T - 2} \right)!\,\,}}} = \frac{{T\left( {T - 1} \right)}}{2}\]
\[\left( {**} \right)\,\,\,\left( { - 5} \right) \cdot \left( { - 6} \right) = 30\,\,\,\,\, \Rightarrow \,\,\,\,T = - 5\,\,\,\,\left( {{\text{not}}\,\,{\text{viable}}} \right)\]
(In words: we have a second-degree equation in the variable T, and we found the two real distinct roots of it: 6 and -5. From the fact that -5 is negative, our answer is really 6.)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.