M45-22

Question

Bob is organizing his stamp collection by placing all 56 of his stamps into several albums. Is there at least one album that contains two or more stamps of the same color?

(1) Each stamp in the collection is exactly one of the following six colors: red, blue, green, yellow, orange, or violet.
 
(2) Bob places the stamps into 9 albums, with each stamp placed in exactly one album.

Bunuel · Answer

Official Solution:

Bob is organizing his stamp collection by placing all 56 of his stamps into several albums. Is there at least one album that contains two or more stamps of the same color? 
 
(1) Each stamp in the collection is exactly one of the following six colors: red, blue, green, yellow, orange, or violet.
 
If there are 56 albums, we could place one stamp in each, and the answer would be NO. But if there are only 2 albums, one of them could contain two red stamps, giving a YES answer. Not sufficient.
 
(2) Bob places the stamps into 9 albums, with each stamp placed in exactly one album. 
 
If all 56 stamps are of different colors, no album will contain two of the same color. But if the stamps are only of two colors, then at least one album must contain two of the same color. Not sufficient.
 
(1)+(2) We can distribute 6 different color stamps (1 red, 1 blue, 1 green, 1 yellow, 1 orange, and 1 violet) to album 1, album 2, and so on until album 9. This covers 6 * 9 = 54 stamps. Each of the remaining 2 stamps must go into an album that already has a stamp of the same color. So in any case, at least one album will contain two or more stamps of the same color. Sufficient.
 
Answer: C

NikunjC · Answer

I did not quite understand the solution. Hello,
I get how you came to the solution, but i am having difficulty in understanding the assumption that there will be equal number of stamps of each color and that the albums will only contain stamps of one colour.

It could be the case that a album contains all different colors right.

hence i chose the option e

Bunuel · Answer

Your confusion comes from misreading the logic. The solution does not assume that each album contains only one color or that stamps are evenly divided by color. It simply uses the maximum possible spread. With 6 colors and 9 albums, even if you try to distribute the stamps as evenly as possible (to avoid color repeats), you can only place 6 * 9 = 54 unique color–album combinations. Since there are 56 stamps, 2 extra stamps must repeat a color already placed somewhere, forcing at least one album to have two stamps of the same color. Check alternative solutions here: https://gmatclub.com/forum/gmat-club-ol ... 47274.html

AbantikaDeka · Answer

I don’t quite agree with the solution. We don't know the quantity of each colour what if we have 12 reds,1 blue and different number of other colours, then this answer will not hold true

Bunuel · Answer

You are missing a point and not reading the question and solution carefully.

Having 12 reds does  not  break the logic. It makes duplicates even more unavoidable. If 6 colors already force a duplicate by stamp 55, then having  more  of any color only forces a duplicate earlier. How do you suggest placing 12 red stamps into 9 albums such that no album contains more than one red stamp?

M45-22

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