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04 Jul 2025, 09:13
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The Question: Is there at least one album that contains two or more stamps of the same color?
This is a "Yes/No" question. We need to determine if we can definitively answer "Yes" or "No".
We have 56 stamps in total.
Statement (1): Each stamp in the collection is exactly one of the following six colors: red, blue, green, yellow, orange, or violet.
This tells us there are 6 distinct colors.
It does not tell us anything about how many albums Bob uses, or how the stamps are distributed among the albums.
For example: If he uses 1 album, he'd definitely have multiple stamps of the same color. If he uses 60 albums, it's possible each stamp is in its own album (but he only has 56 stamps anyway, so this case is impossible).
Is (1) alone sufficient? No.
Statement (2): Bob places the stamps into 9 albums, with each stamp placed in exactly one album.
This tells us there are 9 albums.
It does not tell us anything about the colors of the stamps. We don't know if there are enough unique colors to put one of each color in each album, or if there are only a few colors.
For example: If all 56 stamps were red, then yes, there would be multiple red stamps in one album (since 56 stamps / 9 albums means at least one album has 56/9 = 6.something stamps, all of which would be red).
If there were 56 distinct colors, and he put 6 in each of 8 albums and 8 in the last album, then no album would necessarily have the same color stamp if he distributed them perfectly. But we don't know the number of colors.
Is (2) alone sufficient? No.
Combining Statement (1) and Statement (2):
We have 56 stamps.
We have 9 albums.
We have 6 distinct colors (red, blue, green, yellow, orange, violet).
Now, let's use the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more "pigeons" than "pigeonholes," at least one pigeonhole must contain more than one pigeon.
In this scenario:
Consider each album as a "pigeonhole." There are 9 albums.
Consider the colors within each album as "pigeons."
Let's think about the maximum number of stamps Bob could place in an album without having two stamps of the same color.
Since there are 6 distinct colors, he could place at most 6 stamps in an album, with each stamp being a different color (e.g., 1 red, 1 blue, 1 green, 1 yellow, 1 orange, 1 violet).
If each of the 9 albums contained 6 stamps, each of a different color, he would use 9×6=54 stamps.
However, Bob has 56 stamps.
Since he has 56 stamps and can only place a maximum of 6 distinct colors in each of the 9 albums without repetition (9 albums×6 colors/album=54 possible distinct stamps), the remaining 56−54=2 stamps must go into albums that already contain one stamp of their color.
Therefore, at least two albums will have two stamps of the same color. (More precisely, since 54 stamps can be distributed such that each album has 6 distinct colors, the 55th stamp will cause one album to have a repeated color, and the 56th stamp will cause another, or the same album, to have a repeated color).
This means we can definitively answer "Yes" to the question.
Since both statements together are needed to answer the question, but neither alone is sufficient, the correct GMAT Data Sufficiency option is (C).
The final answer is C
This is a "Yes/No" question. We need to determine if we can definitively answer "Yes" or "No".
We have 56 stamps in total.
Statement (1): Each stamp in the collection is exactly one of the following six colors: red, blue, green, yellow, orange, or violet.
This tells us there are 6 distinct colors.
It does not tell us anything about how many albums Bob uses, or how the stamps are distributed among the albums.
For example: If he uses 1 album, he'd definitely have multiple stamps of the same color. If he uses 60 albums, it's possible each stamp is in its own album (but he only has 56 stamps anyway, so this case is impossible).
Is (1) alone sufficient? No.
Statement (2): Bob places the stamps into 9 albums, with each stamp placed in exactly one album.
This tells us there are 9 albums.
It does not tell us anything about the colors of the stamps. We don't know if there are enough unique colors to put one of each color in each album, or if there are only a few colors.
For example: If all 56 stamps were red, then yes, there would be multiple red stamps in one album (since 56 stamps / 9 albums means at least one album has 56/9 = 6.something stamps, all of which would be red).
If there were 56 distinct colors, and he put 6 in each of 8 albums and 8 in the last album, then no album would necessarily have the same color stamp if he distributed them perfectly. But we don't know the number of colors.
Is (2) alone sufficient? No.
Combining Statement (1) and Statement (2):
We have 56 stamps.
We have 9 albums.
We have 6 distinct colors (red, blue, green, yellow, orange, violet).
Now, let's use the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more "pigeons" than "pigeonholes," at least one pigeonhole must contain more than one pigeon.
In this scenario:
Consider each album as a "pigeonhole." There are 9 albums.
Consider the colors within each album as "pigeons."
Let's think about the maximum number of stamps Bob could place in an album without having two stamps of the same color.
Since there are 6 distinct colors, he could place at most 6 stamps in an album, with each stamp being a different color (e.g., 1 red, 1 blue, 1 green, 1 yellow, 1 orange, 1 violet).
If each of the 9 albums contained 6 stamps, each of a different color, he would use 9×6=54 stamps.
However, Bob has 56 stamps.
Since he has 56 stamps and can only place a maximum of 6 distinct colors in each of the 9 albums without repetition (9 albums×6 colors/album=54 possible distinct stamps), the remaining 56−54=2 stamps must go into albums that already contain one stamp of their color.
Therefore, at least two albums will have two stamps of the same color. (More precisely, since 54 stamps can be distributed such that each album has 6 distinct colors, the 55th stamp will cause one album to have a repeated color, and the 56th stamp will cause another, or the same album, to have a repeated color).
This means we can definitively answer "Yes" to the question.
Since both statements together are needed to answer the question, but neither alone is sufficient, the correct GMAT Data Sufficiency option is (C).
The final answer is C
04 Jul 2025, 09:25
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Bob has 56 stamps in total, each one of six possible colors, placed into 9 albums. If no album had two or more stamps of the same color, then each album could hold at most 6 stamps (one of each color). So, with 9 albums, the maximum number of stamps without repeating a color in any album would be 9 times 6, which is 54 stamps. But Bob has 56 stamps, which is more than 54, so it’s impossible to avoid having at least one album with two or more stamps of the same color. Statement (1) alone doesn’t give the number of albums, and statement (2) alone doesn’t give the number of colors, so neither alone is enough, but together they prove that there must be an album with repeated colors.
Answer: E
Answer: E
04 Jul 2025, 09:38
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even if 6 unique stamps each are segregated into 9 albums we get 54 stamps and remaining 2 for sure are going to be duplicate colours.
this is the best case worst case is there are more duplicate colored stamps, in that case we will have even higher prob to have duplicate colours.
this is the best case worst case is there are more duplicate colored stamps, in that case we will have even higher prob to have duplicate colours.
Bunuel
