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Adriana´s room contains N identical light bulbs with independent switc

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Adriana´s room contains N identical light bulbs with independent switc  [#permalink]

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New post 07 Feb 2019, 10:37
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Question Stats:

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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

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Adriana´s room contains N identical light bulbs with independent switc  [#permalink]

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New post 07 Feb 2019, 11:42
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


\(xC2+xC3+..+xCx=26\)
putting value 5
\(5C2+5C3+5C4+5C5=26\)
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Re: Adriana´s room contains N identical light bulbs with independent switc  [#permalink]

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New post 07 Feb 2019, 11:58
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


keyword: at least two of the light bulbs are switched on

We are given the total ways, we need to find the n => \(nC_x\) = 26

\(xC_2 + xC_3 + xC_4\) = 26

We can try back solving here, Only 5 works perfectly.

\(5C_2\) +\(5C_3\) + \(5C_4\) + \(5C_5\) = 26

B
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Re: Adriana´s room contains N identical light bulbs with independent switc  [#permalink]

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New post 08 Feb 2019, 06:37
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.
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Re: Adriana´s room contains N identical light bulbs with independent switc  [#permalink]

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New post 05 May 2019, 11:32
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Is there any conventional approach to this question (without plugging in)?
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Re: Adriana´s room contains N identical light bulbs with independent switc   [#permalink] 05 May 2019, 11:32
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