fskilnik wrote:
GMATH practice exercise (Quant Class 18)
Adriana´s room contains N identical light bulbs with independent switches, and this room is considered well lighted only if at least two of the light bulbs are switched on. If there are exactly 26 different ways to make Adriana´s room well lighted, what is the value of N?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
Our "official solution" (similar to the previous contributions) is the following:
\(?\,\,\,:\,\,\,N \ge 4\,\,{\mathop{\rm int}} \,\,{\rm{such}}\,\,{\rm{that}}\,\,\,B\left( N \right) = 26\,,\,\,\,{\rm{where}}\,\,\,B\left( N \right) = C\left( {N,2} \right) + C\left( {N,3} \right) + \ldots + C\left( {N,N} \right)\,\,\,\,\,\,\,\left( * \right)\)
\(\left( {\rm{A}} \right)\,\,\,N = 4\,\,\,\,\, \Rightarrow \,\,\,\,B\left( 4 \right)\,\, = \,\,C\left( {4,2} \right) + C\left( {4,3} \right) + C\left( {4,4} \right)\,\, = \,\,{{4 \cdot 3} \over 2} + 4 + 1\,\, = \,\,11\,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle\)
\(\left( {\rm{B}} \right)\,\,\,N = 5\,\,\,\,\, \Rightarrow \,\,\,\,B\left( 5 \right)\,\, = \,\,C\left( {5,2} \right) + C\left( {5,3} \right) + C\left( {5,4} \right) + C\left( {5,5} \right)\,\, = \,\,2 \cdot \left( {{{5 \cdot 4 \cdot 3} \over {3 \cdot 2}}} \right) + 5 + 1\,\, = \,\,20 + 5 + 1\,\,\,\,\,\,\,\left\langle {{\rm{YES}}\,{\rm{!}}} \right\rangle\)
The correct answer is therefore (B).
We follow the notations and rationale taught in the
GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here:
https://gmath.net
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52% (02:02) correct 
