A certain office supply store stocks 2 sizes of self-stick

swerve 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 pacakages 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

The OA is C

Source: GMAT Prep

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 =[spoiler] 16 = C[/spoiler]

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Note: the FCP can be used to solve the majority of counting questions on the GMAT. For more information about the FCP, watch this free video: https://www.gmatprepnow.com/module/gmat-counting?id=775

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Cheers,
Brent

swerve 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 pacakages 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

The OA is C

Source: GMAT Prep

Let’s say the 2 sizes of notepads are small and large. Then, for the small notepads, there are 4 packages of notepads of all the same color (one package for each color) and 4C3 = 4 packages of notepads of three different colors. Thus, for the small notepads, there are a total of 4 + 4 = 8 different packages. Similarly, there are 8 different packages for the large notepads. Thus, there are a total of 8 + 8 = 16 different packages for the 2 sizes of notepads.

Answer: C