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Assembly line

Expert replies
by manik11 » Thu Oct 29, 2015 7:29 am
An auto assembly plant performs six functions with each frame that arrives: add axles, add wheels to the axles, install the windshield to the frame, install the instrument panel, install the steering wheel, and install the interior seating. Once those six tasks are performed, each car goes to a separate building for finishing touches. If these tasks can be arranged along a linear assembly line in any order, except that the axles must be installed before the wheels can be added, how many ways can the assembly line be arranged?

120

240

360

480

720

OA : C
Source : Veritas Prep

According to the official explanation half of the time wheels will come before axles and half of the time axles will come before wheels. I can't visualize this scenario.
Would really appreciate if the experts could break this down for me.
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Source: — Problem Solving |

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by DavidG@VeritasPrep » Thu Oct 29, 2015 7:47 am
According to the official explanation half of the time wheels will come before axles and half of the time axles will come before wheels. I can't visualize this scenario.
Would really appreciate if the experts could break this down for me.
Sometimes it's easier to conceptualize these issues with very simple scenarios. Imagine that attaching the wheels and attaching the axles are the only two procedures that need to be performed. Call attaching the wheels 'W' and call attaching the axles "A." So there are two sequences:

1) wheels first and then axles: WA
2) axles first and then wheels: AW

Clearly half the time wheels come first and half the time axles come first.

Now add one procedure, say, steering wheel. Call it S. Now there will be 3!, or 6 ways to arrange the procedures.

SAW
SWA
WSA
WAS
ASW
AWS

Notice that, again, half the time wheels come first (in red) and half the time axles come first. This makes sense. If there are two scenarios - either wheels must come first or axles must come first - then half the time it will be one scenario and half the time it will be the other.

That will always be true, no matter how many procedures are performed. So if we know that axles must come first, that will eliminate all the scenarios when wheels come first, or half of the total possibilities.
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