Candies
In the box are 12 candies that look the same. Three are filled with nougat, five with nuts, and four with cream. How many sweets must Ivan choose to satisfy himself by selecting two with the same filling?
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Dr. Math
To determine how many sweets Ivan must choose to guarantee that he has at least two with the same filling, we can use the Pigeonhole Principle.
There are three types of fillings: nougat (3), nuts (5), and cream (4). The goal is to ensure that at least two sweets have the same filling.
Consider the worst-case scenario: Ivan picks sweets all with different fillings. Since there are only three distinct fillings, the maximum number of sweets he can pick without having two of the same is three (one of each filling: one nougat, one nuts, and one cream).
However, if he picks a fourth sweet, no matter which filling it is (nougat, nuts, or cream), it will necessarily match one of the fillings he already has. Therefore, he will have at least two sweets with the same filling.
Thus, Ivan must choose 4 sweets to be sure that at least two have the same filling.
There are three types of fillings: nougat (3), nuts (5), and cream (4). The goal is to ensure that at least two sweets have the same filling.
Consider the worst-case scenario: Ivan picks sweets all with different fillings. Since there are only three distinct fillings, the maximum number of sweets he can pick without having two of the same is three (one of each filling: one nougat, one nuts, and one cream).
However, if he picks a fourth sweet, no matter which filling it is (nougat, nuts, or cream), it will necessarily match one of the fillings he already has. Therefore, he will have at least two sweets with the same filling.
Thus, Ivan must choose 4 sweets to be sure that at least two have the same filling.
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