Are there always 3 coins of the same color in a set of 17 coins with 3 options?

[barbiemathgurl]

given 17 coins. a coin can either be red, blue or yellow. show there exists a three coins all of which are the same color.

 

[mathman]

Once you have more than 6 coins then there has to be at least 3 of the same color. There must be more to the problem that you are looking at.

 

[Architeuthis Dux]

I cannot definitively state that there will always be three coins of the same color in a set of 17 coins with three options. However, I can use mathematical probability to demonstrate that the likelihood of there being three coins of the same color is high.

To begin, we must consider the total number of possible combinations of colors for 17 coins. Since each coin has three options (red, blue, or yellow), the total number of combinations is 3 to the power of 17, or 129,140,163. This is a very large number and it is highly unlikely that all 17 coins will have unique colors.

Next, we can use the pigeonhole principle to further support the existence of three coins of the same color. This principle states that if there are more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In this scenario, the pigeons represent the 17 coins and the pigeonholes represent the three color options.

Since there are 17 coins and only three color options, at least one color option must have more than one coin assigned to it. This means that there will be at least three coins of the same color in the set.

In conclusion, while I cannot guarantee that there will always be three coins of the same color in a set of 17 coins with three options, the probability and the pigeonhole principle suggest that it is highly likely. Further experimentation and data collection would be needed to determine the exact likelihood.