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Permutation/Combination - Simple one

by J N » Sat Jun 01, 2013 8:36 am
A certain car dealership offers its newest model in ten exterior and ten interior colors. If the ten interior colors are identical to the ten exterior colors, and the dealership sells every pair of colors except those that would result in a car with an identically colored interior and exterior, how many different color combinations are possible?

45
81
90
10!/2!
10!

when they say different color combinations does that mean fro instance red exterior and blue interior and blue exterior and red interior are the same? colors are obviously on different parts of car but the combinations of blue and red is only 1


FYI - answer is 90 so i assume they meant 10 *9 but could easily see how someone would pick (10*9)/2

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by Brent@GMATPrepNow » Sat Jun 01, 2013 9:31 am
J N wrote:A certain car dealership offers its newest model in ten exterior and ten interior colors. If the ten interior colors are identical to the ten exterior colors, and the dealership sells every pair of colors except those that would result in a car with an identically colored interior and exterior, how many different color combinations are possible?

45
81
90
10!/2!
10!

when they say different color combinations does that mean fro instance red exterior and blue interior and blue exterior and red interior are the same? colors are obviously on different parts of car but the combinations of blue and red is only 1


FYI - answer is 90 so i assume they meant 10 *9 but could easily see how someone would pick (10*9)/2
The correct answer (90) indicates that a car with red exterior and blue interior is different from a car with blue exterior and red interior.

Here's one approach:

Take the task of choosing colors break it into stages.

Stage 1: Choose the exterior color
Since there are 10 color choices, we can complete this stage in 10 ways

Stage 2: Choose the interior color
At this point, we can't choose the color that was used for the exterior.
So, we have 9 color options remaining, which means we can complete this stage in 9 ways

By the Fundamental Counting Principle (FCP) we can complete the 2 stages (and thus design the car) in (10)(9) ways ([spoiler]= 90 ways[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775
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by Trueindian » Mon Jun 03, 2013 12:45 am
J N wrote:A certain car dealership offers its newest model in ten exterior and ten interior colors. If the ten interior colors are identical to the ten exterior colors, and the dealership sells every pair of colors except those that would result in a car with an identically colored interior and exterior, how many different color combinations are possible?

45
81
90
10!/2!
10!

when they say different color combinations does that mean fro instance red exterior and blue interior and blue exterior and red interior are the same? colors are obviously on different parts of car but the combinations of blue and red is only 1


FYI - answer is 90 so i assume they meant 10 *9 but could easily see how someone would pick (10*9)/2
When they say different color combinations - Red Exterior/Blue Interior and Blue Ext/Red Int are 2 different combinations.

Thus, a simple way of solving can be:

The exterior color can be chosen in 10 different ways. For each exterior color the interior color can be chosen in 9 different ways (Excluding the 1 color, which is used as exterior color)

Hence, the total number of ways will be = The number of ways Exterior color can be chosen * The number of ways interior color can be chosen = 10 * 9 = 90

Cheers,
True

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by sana.noor » Sat Jul 20, 2013 11:41 am
Brent, what if we add a restriction in this question that says that "Order of colors isnt important" than is the answer 45?
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by jitsy » Sat Jul 20, 2013 12:54 pm
Hi Sana, before Brent answers that question, I'd like to understand for my own self, what you mean by order of colors here.
sana.noor wrote:Brent, what if we add a restriction in this question that says that "Order of colors isnt important" than is the answer 45?
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by Brent@GMATPrepNow » Sat Jul 20, 2013 3:07 pm
sana.noor wrote:Brent, what if we add a restriction in this question that says that "Order of colors isnt important" than is the answer 45?
Yes, the answer would then be 45.
The question would simplify to choosing two colors from ten colors. Since the order of the selected colors does not matter (e.g., a car with red exterior and blue interior is THE SAME as a car with blue exterior and red interior), we can use combinations.
We can select 2 colors from 10 colors in 10C2 ways (45 ways)

Aside: If anyone is interested, we have a free video on calculating combinations (like 10C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

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by sana.noor » Sun Jul 21, 2013 1:22 am
jitsy wrote:Hi Sana, before Brent answers that question, I'd like to understand for my own self, what you mean by order of colors here.
sana.noor wrote:Brent, what if we add a restriction in this question that says that "Order of colors isnt important" than is the answer 45?
in permutation "order of selected thing" matter but in combination order of selected thing doesnt matter. for example: AB and BA is counted two different things in permutation because order is important. AB and BA is considered two distinct things. but in combination AB and BA is same thing because order isnt important. if u have selected AB then you cant select BA.
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by jitsy » Sun Jul 21, 2013 2:29 am
Hi Sana, thanks. Sorry I had meant I do understand the concept of order in P&C but could not relate it to this question since I was thinking of it in the literal 'order' sense when its actually an interior exterior swap. But I realised what you meant when Brent explained. Thanks again
sana.noor wrote:
jitsy wrote:Hi Sana, before Brent answers that question, I'd like to understand for my own self, what you mean by order of colors here.
sana.noor wrote:Brent, what if we add a restriction in this question that says that "Order of colors isnt important" than is the answer 45?
in permutation "order of selected thing" matter but in combination order of selected thing doesnt matter. for example: AB and BA is counted two different things in permutation because order is important. AB and BA is considered two distinct things. but in combination AB and BA is same thing because order isnt important. if u have selected AB then you cant select BA.
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by sandmandreams » Thu Aug 15, 2013 10:20 pm
I computed by getting all possible combinations 10 x 10 = 100; and then subtracting the same colored pairs,which is 10

100-90 = 10

I think that would be clearer for you to avoid confusion.

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by j_shreyans » Thu May 14, 2015 8:02 am
Hi Brent ,

One doubt, why did you do till stage2 as we have 10 interior colors are identical to the ten exterior colors and we have to make different combination so should not be 10X9X8X7X.......X1 = 10!

Please correct me if i am wrong.

Thanks,

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by Brent@GMATPrepNow » Thu May 14, 2015 8:12 am
j_shreyans wrote:Hi Brent ,

One doubt, why did you do till stage2 as we have 10 interior colors are identical to the ten exterior colors and we have to make different combination so should not be 10X9X8X7X.......X1 = 10!

Please correct me if i am wrong.

Thanks,

Shreyans
Hi Shreyans,

Once we have selected a color for the exterior (which we can do in 10 ways), we must select a color for the interior. Since we can't repeat the same color, we now have only 9 colors to choose from. That's it.

Aside: 10! = 3.6 million. If you were to start listing all of the possible 2-color combinations, you'd soon realize that there can't be 3.6 million possibilities.

Cheers,
Brent
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