Of the 5 distinguishable wires that lead into an apartment, 2 are for

Question

Of the 5 distinguishable wires that lead into an apartment, 2 are for cable television service, and 3 are for telephone service. Using these wires, how many distinct combinations of 3 wires are there such that at least 1 of the wires is for cable television?

A. 6
B. 7
C. 8
D. 9
E. 10

Show Spoiler

My Approach is as follows :
Since there are 2 cables wires and 3 TV wires ....and there has to be a minimum of 1 cable wire in the combination of three wires so we have 2 cases :

Case 1 - Choosing 1 cable wire out of 2 and choosing 2 TV wires out of 3 ...This is 2C1 * 3C2
OR
Case 2 - Choosing 1 cable wire wire in 1st place, then choosing 1 cable wire for 2nd place and finally choosing 1 TV wires out of 3 ...This is 1 * 1 * 3C1
Thus answer is case 1 OR Case 2

This is 2C1 * 3C2 + 1 *1 * 3C1 = 6 + 3 = 9 ...
Please correct me if I am wrong here.

Show Answer

Official Answer

D

blink005 · Accepted Answer

Your approach is correct.
There is another approach
There are 5 wires in all
2 cable and 3 telephone service

To find the required combinations you can subtract the no. of combinations of all telephone wires from all possible combinations.

Required no. of combinations= 5C3-3C3=10-1=9

Hope this helps

vishalrastogi · Answer

Hi Bunuel,

Have solved this by using this method : 3C2 X 2C1 + 3C1 X 2C2 = 9.

Is this correct ?

Bunuel · Answer

Yes, it is. It's a direct approach in contrast to  {total}-{restriction}  approach used in the posts above.

vishalrastogi · Answer

Thanks Bunuel. I always have my fingers crossed when I solve combinations.

chetan2u · Answer

Hi,

In these Q, where atleast is used, the best is to find none and then subtract from total..
here total ways= 5C3=5!/3!2!=10..
ways where no cable television is used=3c3=1..
therefore ways containing atleast one wire of cable tele=10-1=9..
ans D

hdwnkr · Answer

Thanks to the GMAT Club Maths Book for explaining Combinations. Hope, I've done justice to the same (with a correct answer)

Given: 
Total number of wires = 5
Number of cable wires = 2
Number of telephone wires = 3

To find: 
Number of combination which has at least one cable wires

Solution: 
No of ways of selecting  'at least'  1 cable wire means, we can select more than one as well. The minimum we can select is one and the maximum we can select, given the constraints that 3 wires need to be selected in total and there are 2 cable wires, is 2
 
Since it is a combination of wires, the arrangement is not important

Approach 1:

Number of ways of selecting at least one cable wire in a selection of 3 wires from 5 wires =  Selection 1  (Number of ways of selecting one cable wire and two telephone wires )+  Selection 2  (Number of ways of selecting two cable wires and 1 telephone wire)

Selection 1
Number of ways of selecting one cable wire = 2C1 = 2
Number of ways of selecting 2 telephone wires = 3C2 = 3
Total = 2C1 * 3C2 = 6 ( m ways of doing something and n ways of doing something else together give m*n ways of doing - the holy grail rule in Combinatorics)

Selection 2
Number of ways of selecting one cable wire = 2C2 = 1
Number of ways of selecting 2 telephone wires = 3C1 = 3
Total = 2C2 * 3C1 = 3 ( m ways of doing something and n ways of doing something else together give m*n ways of doing - the holy grail rule in Combinatorics)

Selection 1 + Selection 2 = 9 ways of selecting 3 wires out of 5 such that at least one is a cable wire

Approach 2

Number of ways of selecting 3 wires out of 5 such that at least one is a cable wire =  Selection X  (Total number of ways of selecting 3 wires from the 5) -  Selection Y  (total ways of selecting 3 wires such that none is a cable i.e all the three are telephone wires)

Total number of ways of selecting 3 wires out of 5 = 5C2 = 10
Number ways of selecting 3 wires such that none is a cable i.e all the three are telephone wires = 3C3 ( 3 telephone wires and we are selecting all the three at once) = 1

Selection X - Selection Y = 9

Answer is Option D

GMAT Club Math Book  https://gmatclub.com/forum/gmat-math-book-in-downloadable-pdf-format-130609.html

Kelzie01 · Answer

So I suck at combinations/permutations/whichever one this question was. When there aren't many possibilities I have better luck just listing the combinations.

CTT
TCT
TTC
CCT
TCC
CTC

I am apparently missing three though? According to the other explanations which seem a lot more correct then what I did...

chetan2u · Answer

Hi,
why you are going wrong is because you are missing out on an important word.. distinguishable ..
so take C as C1 and C2... T as T1,T2,T3..
now try, you will get the answer

Kelzie01 · Answer

Hmm...it seems that will give me way more than 9? Way,  way  more. Even just adding C1 and C2 to the first 3 gets me to 9-not even considering that there is T1, T2...

C1TT
TC1T
TTC1
C2TT
TC2T
TTC2
CCT
TCC
CTC

chetan2u · Answer

hi,
one of the C1 and C2 is must and rest three are T1,T2,T3..
the ways are 
C1,C2,T1
C1,C2,T2
C1,C2,T3
C1,T1,T2
C1,T3,T2
C1,T1,T3
C2,T1,T2
C2,T1,T3
C1,T3,T2..
there are 9 ways..
and any one above is just one way.. 
you can't take C1,C2,T1 as 
C1,C2,T1..
C2,C1,T1
T1,C2,C1
you are picking up three, order does not matter as  The Q is about combinations ..
hope it is clear now..

GMATinsight · Answer

Total ways of choosing 3 wires out of 5 = 5C3 = 10

(Unfavorable outcomes)   ways of choosing 3 such that none of them is for cable = 3C3 (there are 3 wires without cable) = 1

Favorable outcomes = 10-1 = 9

Answer: Option D

Of the 5 distinguishable wires that lead into an apartment, 2 are for

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Of the 5 distinguishable wires that lead into an apartment, 2 are for

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards