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29 Jul 2024, 02:30
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Emily has a box containing 15 blue contact lenses, 20 green contact lenses, 25 brown contact lenses, and 30 gray contact lenses. She has a 6-day vacation and wants to take 6 matching-color pairs with her (for example, 6 pairs of blue lenses OR 2 pairs of green lenses, 3 pairs of gray lenses, and 1 pair of brown lenses OR 1 pair of blue lenses, 1 pair of green lenses, 2 pairs of brown lenses, and 2 pairs of gray lenses, and so on.). What is the least number of contact lenses that Emily must randomly select from the box to guarantee that her selections contain at least 6 pairs of contact lenses?
A. 12
B. 13
C. 14
D. 15
E. 16
A. 12
B. 13
C. 14
D. 15
E. 16
29 Jul 2024, 02:30
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Official Solution:
Emily has a box containing 15 blue contact lenses, 20 green contact lenses, 25 brown contact lenses, and 30 gray contact lenses. She has a 6-day vacation and wants to take 6 matching-color pairs with her (for example, 6 pairs of blue lenses OR 2 pairs of green lenses, 3 pairs of gray lenses, and 1 pair of brown lenses OR 1 pair of blue lenses, 1 pair of green lenses, 2 pairs of brown lenses, and 2 pairs of gray lenses, and so on.). What is the least number of contact lenses that Emily must randomly select from the box to guarantee that her selections contain at least 6 pairs of contact lenses?
A. 12
B. 13
C. 14
D. 15
E. 16
To guarantee that Emily has at least 6 pairs of contact lenses of the same color, we need to consider the worst-case scenario. What is the maximum number of contact lenses Emily can select without having 6 pairs of contact lenses?
She could pick 5 pairs of contact lenses, which means 10 lenses in total. Then, since there are four different colors of lenses, she can pick 1 lens from each color, making a total of 14 lenses. At this point, Emily would still not have 6 pairs of contact lenses of the same color. However, the next contact lens, the 15th, regardless of its color, would complete a pair, giving her 6 pairs of lenses. For example, she could have 5 pairs of blue lenses (10 blue lenses), 1 blue contact lens, 1 green contact lens, 1 brown contact lens, and 1 gray contact lens. The next lens, regardless of its color, would complete a pair, ensuring she has 6 pairs of contact lenses.
Hence, for any selection with fewer than 15 lenses, we could have fewer than 6 pairs. However, with any selection of 15 or more lenses, it is guaranteed to have at least 6 pairs of the same colored lenses.
Thus, the least number of contact lenses that Emily must select to guarantee that her selections contain at least 6 pairs of contact lenses is 15.
Answer: D
Emily has a box containing 15 blue contact lenses, 20 green contact lenses, 25 brown contact lenses, and 30 gray contact lenses. She has a 6-day vacation and wants to take 6 matching-color pairs with her (for example, 6 pairs of blue lenses OR 2 pairs of green lenses, 3 pairs of gray lenses, and 1 pair of brown lenses OR 1 pair of blue lenses, 1 pair of green lenses, 2 pairs of brown lenses, and 2 pairs of gray lenses, and so on.). What is the least number of contact lenses that Emily must randomly select from the box to guarantee that her selections contain at least 6 pairs of contact lenses?
A. 12
B. 13
C. 14
D. 15
E. 16
To guarantee that Emily has at least 6 pairs of contact lenses of the same color, we need to consider the worst-case scenario. What is the maximum number of contact lenses Emily can select without having 6 pairs of contact lenses?
She could pick 5 pairs of contact lenses, which means 10 lenses in total. Then, since there are four different colors of lenses, she can pick 1 lens from each color, making a total of 14 lenses. At this point, Emily would still not have 6 pairs of contact lenses of the same color. However, the next contact lens, the 15th, regardless of its color, would complete a pair, giving her 6 pairs of lenses. For example, she could have 5 pairs of blue lenses (10 blue lenses), 1 blue contact lens, 1 green contact lens, 1 brown contact lens, and 1 gray contact lens. The next lens, regardless of its color, would complete a pair, ensuring she has 6 pairs of contact lenses.
Hence, for any selection with fewer than 15 lenses, we could have fewer than 6 pairs. However, with any selection of 15 or more lenses, it is guaranteed to have at least 6 pairs of the same colored lenses.
Thus, the least number of contact lenses that Emily must select to guarantee that her selections contain at least 6 pairs of contact lenses is 15.
Answer: D
07 Sep 2024, 23:46
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate.
