Car with seven options

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Jan 18, 2009 9:13 am

A certain make of car has the following seven options: air conditioning, sunroof, DVD player, CD changer, leather upholstery, tinted windows and cruise control. A person ordering this make of car has how many different combinations from which to choose?
(A) 49
(B) 56
(C) 98
(D) 128
(E) 840

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Source: Beat The GMAT — Problem Solving | Sun Jan 18, 2009 9:13 am

DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Sun Jan 18, 2009 9:16 am

I'd go with 7! = 840, but I may be very wrong...

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Jan 18, 2009 9:33 am

Sorry, it's not E.

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Sun Jan 18, 2009 9:37 am

You're right... I'm so dumb... It's 0C7 + 1C7 + ... + 7C7 = 2^7 = 128. Sorry...

Last edited by DanaJ on Sun Jan 18, 2009 9:45 am, edited 1 time in total.

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by Brent@GMATPrepNow » Sun Jan 18, 2009 9:42 am

128 it is.
One option is to consider 8 different cases (get zero options, get 1 option, get 2 options, get 3 options and so on). So we get the total combinations as 7C0 + 7C1 + 7C2 + . . . + 7C6 + 7C7
This is a lot of work.

The other strategy is to break this task into seven stages:
1) Choose whether or not to get air conditioning, sunroof, DVD player
2) Choose whether or not to get air the sunroof
. . .
7) Choose whether or not to get cruise control

Each stage can be accomplished in 2 ways.
So, by the Fundamental Counting Principle, the total number of ways to accomplish all 7 stages is 2x2x2x2x2x2x2 = 128

In general, the sum nC0 + nC1 + nC2 + . . . nCn = 2^n

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Sun Jan 18, 2009 9:45 am

Hmm... I messed up the C's over there... because this is not standard notation in my country... Over here it's smth like both numbers are in front of the C, the bigger one on the bottom and the smaller one on top... I guess I should have written 7C0 + 7C1 + etc...

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sana.noor: Legendary Member; Posts: 512; Joined: Mon Jun 18, 2012 11:31 pmFollowed by: 20 members; Thanked: 42 times

by sana.noor » Tue Jul 30, 2013 11:41 pm

D it should be as we have 2 options for every 7 options 2.2.2.2.2.2.2 = 128.

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