Ben has 30 pencils in a box. Each of the pencils is one of 5 different colors, and there are 6 pencils of each color. If Ben selects pencils one at a time from the box without being able to see the pencils, what is the minimum number of pencils that he must select in order to ensure that he selects at least 2 pencils of each color?
A. 24
B. 25
C. 26
D. 27
E. 28
Answer:
This happens to be a GRE question, but it is actually a replica or a common brain teaser (see Martin Gardner’s Colored Socks problem). The easiest way to solve this problem is to imagine how many pencils you could select and NOT get 2 of each color. Well, you could select all pencils of one color, all of another color, and so on. If you selected all of the pencils of 4 of the colors, you would have 24 pencils (6 of each of the 4 colors). You’d then have 6 pencils remaining, all of the final color. If you select 2 more pencils, you’d then be guaranteed to have 2 of each color. So the answer is C: 26.
Of course, if you ran this experiment, it is likely that you’d select 2 pencils of each color before selecting 26, but the question asks how many Ben must select to ensure that he selects 2 of each color. Since it’s possible that he could select 25 and NOT have 2 of each color, 26 is the least number that must be selected to ensure that he has 2 of each color.