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GMAT PREP PS question

by alex.gellatly » Tue Apr 17, 2012 4:55 am

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A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: blue, green, yellow, or pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of
3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?

(A) 6
(B) 8
(C) 16
(D) 24
(E) 32

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by Anurag@Gurome » Tue Apr 17, 2012 5:00 am
alex.gellatly wrote:A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: blue, green, yellow, or pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of
3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?

(A) 6
(B) 8
(C) 16
(D) 24
(E) 32

Thanks
Number of notepads of the same color = 4 (blue, green, yellow, pink)
Since there are two different sizes, so total number of notepads for the same color = 4 * 2 = 8

We have to choose 3 different colors from 4, so notepads of different colors = 4C3 = 4
Since there are two different sizes, so total number for the different color = 4 * 2 = 8

Therefore, required number of packages = 8 + 8 = 16

The correct answer is C.
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by aneesh.kg » Tue Apr 17, 2012 5:04 am
Since there is too much information in the question, we can start by making a few symbols to get a better clarity into the question as follows:

Let B, G, Y, P be the big-sized notepads.
Let b, g, y, p be the small-sized notepads.

For Three notepads of same size and same colour we can have a package containing 3 of each of B, G, Y, P, b, g, y and p. such as (B,B,B), (y,y,y), etc. That's a total of 8 packages.

+

For 3 notepads of the same size and of 3 different colors, you can selected three from the big-sized one or 3 from the small-sized ones = 4C3 + 4C3 = 8

8 + 8 = 16

(C) is the answer.
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by ronnie1985 » Tue Apr 17, 2012 9:38 am
2C1*4C1+4C3*2C1 = 8 + 8 = 16
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by amyhussein » Wed Mar 13, 2013 11:51 am
Number of notepads of the same color = 4 (blue, green, yellow, pink)
Since there are two different sizes, so total number of notepads for the same color = 4 * 2 = 8

We have to choose 3 different colors from 4, so notepads of different colors = 4C3 = 4
Since there are two different sizes, so total number for the different color = 4 * 2 = 8
i understand till here, why did we do after that 8 +8?[/quote]
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by Anurag@Gurome » Wed Mar 13, 2013 12:03 pm
amyhussein wrote:i understand till here, why did we do after that 8 +8?
Because the question said "... the store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of 3 different colors"

Those are two different scenarios.
For each of them, 8 different packages are possible.
Hence, total number of possible packages = (8 + 8) = 16

Hope that helps.
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by amyhussein » Wed Mar 13, 2013 11:31 pm
I am a bit confused,
How. An we choose 3 packs with same size and colour if already each size contains total of 4 notes each in different colors. My understanding is that there are no multiple notes with same colour within the same size. What am I missing here
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by Anju@Gurome » Thu Mar 14, 2013 12:22 am
amyhussein wrote:I am a bit confused,
How. An we choose 3 packs with same size and colour if already each size contains total of 4 notes each in different colors. My understanding is that there are no multiple notes with same colour within the same size. What am I missing here
The problem said only the following things,
  • There are notepads of two different size, say large and small.
    Each size has four different colors: blue, green, yellow, or pink.
There is nothing about number of notepads.
In fact, there are multiple number of notepads of same color and same size. Otherwise how the store will pack the notepads in packages that contain 3 notepads of the same size and the same color?

Your mistake is you are assuming that "each size contains total of 4 notes each in different colors"
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Re: GMAT PREP PS question

by Brent@GMATPrepNow » Sat Apr 16, 2022 5:58 am
alex.gellatly wrote: ↑
Tue Apr 17, 2012 4:55 am
 A certain office supply store stocks 2 sizes of self-stick notepads, each in 4 colors: blue, green, yellow, or pink. The store packs the notepads in packages that contain either 3 notepads of the same size and the same color or 3 notepads of the same size and of
3 different colors. If the order in which the colors are packed is not considered, how many different packages of the types described above are possible?

(A) 6
(B) 8
(C) 16
(D) 24
(E) 32

Thanks
There are two different cases to consider:
1) All 3 pads the same color
2) The 3 pads are 3 different colors

Case 1: All 3 pads the same color
Take the task of packaging pads and break it into stages.

Stage 1: Select a size
There are 2 possible sizes, so we can complete stage 1 in 2 ways.

Stage 2: Select 1 color (to be applied to all 3 pads)
There are 4 possible colors from which to choose, so we can complete stage 2 in 4 ways.

By the Fundamental Counting Principle (FCP) we can complete the two stages in (2)(4) ways (= 8 ways)


Case 2: The 3 pads are 3 different colors
Take the task of packaging pads and break it into stages.

Stage 1: Select a size
There are 2 possible sizes, so we can complete stage 1 in 2 ways.

Stage 2: Select 3 different colors
There are 4 possible colors, and we must choose 3 of them.
Since the order of the selected colors does not matter, we can use combinations.
We can select 3 colors from 4 colors in 4C3 ways (4 ways), so we can complete stage 2 in 4 ways.

By the Fundamental Counting Principle (FCP) we can complete the two stages in (2)(4) ways (= 8 ways)


So, both cases can be completed in a total of 8 + 8 ways = 16

Answer: C
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