An auto assembly plant performs six functions with each frame that arr

Question

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?

A. 120
B. 240
C. 360
D. 480
E. 720

Kudos for a correct solution.

A. 120
B. 240
C. 360
D. 480
E. 720

Bowtye · Accepted Answer

C) 360

short way: there are 6! ways to do the six tasks. Half will have wheels before axles and half will have axles before wheels. So we want 6!/2 -> 720/2 = 360

Bunuel · Answer

VERITAS PREP OFFICIAL SOLUTION:

C. In this permutations problem (the order does matter - that's what they're asking about!), there are six items to be arranged in order, so the "unrestricted" permutation would be 6! = 720. But there's a catch - since wheels cannot come before axles, any permutation that would put wheels before axles does not work. How do you find that restriction? It's actually a bit simpler than you might think - if you're considering all possibilities, half the time axles will come before wheels (acceptable) and half the time wheels will come before axles (unacceptable). So take away the unacceptable half of the 720 arrangements and 360 are left.

KarishmaB · Answer

Check out the blog posts on the link given in my signature below.

shriramvelamuri · Answer

I like the Wofford09 answer but for a lengthy solution, here you go...

If Axle can be first job, then rest five job can arranged in 5! ways.

If the Axle is the second job, the Wheel can be done in 4 ways and the remaining 4 jobs can be arranged in 4!.

If the Axle is the third job, the wheel can be done in 3 ways and the reamining 4 jobs can be arranged in 4!.

If the Axle is the second job, the wheel can be done in 2 ways and the reamining 4 jobs can be arranged in 4!.

.....

Finally total permuations.., 5! + (4+3+2+1) 4!= 360 ways.

As soon as I saw the question, I started working on it.... I presume I have a slow a bit to be able to do what wooford09 Did.

Bowtye · Answer

I think knowing the lengthy solution is also very important (and a very good exercise to do). Kudos

KS15 · Answer

Total number of ways in which the two specific functions can be selected out of the 6 functions=6C2
Remaining 4 functions can be arranged in 4! ways.
Answer=6C2*4!=360 (C)

Nadiuska · Answer

Hi there,

I'm not able to quite see why is half before and half after. I understand it, but I can not see it!

Can anyone explain?

Thanks

KarishmaB · Answer

Responding to a pm:

Note that axles have to put before wheels but not necessarily immediately before. What I mean to say is that

axles, wheels, windshield, instrument panel, steering wheel, interior seating
is possible and so is
axles, windshield, instrument panel, steering wheel, interior seating, wheels
and also
axles, windshield, instrument panel, wheels, steering wheel, interior seating
etc

The point is, as long as axles come sometime before wheels, it is fine.
When you club them together in a group, you are only allowing case 1.

That is the problem.

NarlankaVijay · Answer

The mistake I did was taking 
A,W,_,_,_,_ . Now I took A,W as one block & rest others as separate blocks. This gives 5! as answer. But the logic we are missing is the question didn’t say both A,W should happen immediately after one another. It just said that W should happen after A.

So all the possible combinations are:
A,W,_,_,_,_
A,W,_,_,_,_
A,W,_,_,_,_
A,W,_,_,_,_
A,W,_,_,_,_
_,A,W,_,_,_
_,A,_W,_,_
.
.
.
.
_,_,_,_,A,W

If Axle can be first job, then rest five job can arranged in 5! ways.

If the Axle is the second job, the Wheel can be done in 4 ways and the remaining 4 jobs can be arranged in 4!.

If the Axle is the third job, the wheel can be done in 3 ways and the reamining 4 jobs can be arranged in 4!.

If the Axle is the second job, the wheel can be done in 2 ways and the reamining 4 jobs can be arranged in 4!.

.....

Finally total permuations.., 5! + (4+3+2+1) 4!= 360 ways.

Posted from my mobile device

wizardofoddz · Answer

Total number of ways in which 6 functions can be arranged = 6!

This will have only two types of scenarios involving installation of axles and fitting of wheels.

1. The former can happen before the latter.
2. The latter can happen before the former.

In fact, the number of possible arrangements of both is equal to each other.

Therefore, number of acceptable ways = 6!/2 = 360 ways.

nikitathegreat · Answer

why are we doing 6C2 - how are we ensuring through this that Axles will come before wheels? 4! is something that I can understand where the remaining places can be arranged in 4! ways.

karishma  - Please help  GMATNinja

Krunaal · Answer

In the solution 6C2 * 4!, we are selecting two empty spots on the assembly line using 6C2 where we first place A and then W, and then in the 4 empty spaces we can place the functions in 4! ways. So 6C2 is  not the selection of functions, but of the empty spots on the assembly line . Hope it helps!

nikitathegreat · Answer

Thanks for the response. Also, wanted to understand that 6C2 also takes care of the arrangement - i.e axles should come before the wheels? I thought it was just a way of selecting 2 spots and then doing 4! where the remaining 4 slots can be filled in whatever way.

Krunaal · Answer

Yes, since we select two spots and we know that there is only 1 way to fill those two spots viz. first A and then W, we essentially are doing 6C2 * 1 which is just 6C2

nikitathegreat · Answer

Does 6C2 also takes into account the different arrangements? While we know that A will come before W but there can be different cases for the same AXXWXX or AXXXXWX, etc?

Krunaal · Answer

Yes, since we are selecting two empty spots on the assembly line, those two spots can be anywhere which includes all the possible cases.

An auto assembly plant performs six functions with each frame that arr

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