Jar of Jelly Beans: 20 of Each Color, How Many to Pick?

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Discussion Overview

The discussion revolves around a combinatorial problem involving a jar of jelly beans of different colors. Participants explore how many jelly beans must be selected to ensure that at least three jelly beans of the same color are chosen. The scope includes mathematical reasoning and application of the pigeonhole principle.

Discussion Character

  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant questions the teacher's answer of 9, suggesting that the problem may involve selecting 2 of each color.
  • Another participant explains that after drawing 8 jelly beans, it is possible to have 2 of each color, and the next draw must yield a third jelly bean of one color.
  • A further analysis indicates that if only three colors are present after 8 draws, at least one color must have 3 jelly beans, reinforcing the need for the 9th draw.
  • One participant applies the pigeonhole principle, stating that the worst-case scenario after 8 draws would still necessitate a third jelly bean of one color with the 9th draw.
  • Another participant provides a general formula for determining the minimum number of draws needed to guarantee at least k jelly beans of any color, specifically applying it to the current scenario with 4 colors and 3 jelly beans.

Areas of Agreement / Disagreement

Participants generally agree on the conclusion that 9 jelly beans must be drawn to guarantee at least 3 of the same color, but there is some uncertainty regarding the initial interpretation of the problem and the teacher's answer.

Contextual Notes

There is a reliance on the pigeonhole principle and combinatorial reasoning, and assumptions about the distribution of jelly beans are made throughout the discussion.

yakin
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A jar contains red, blue, orange and green jelly beans. There are 20 of each color. How many jelly beens must be selected to guarantee that at least 3 are the same color?
The teacher got 9 as answer, but i do not know how she got this. Anybody would like to help please?
 
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yakin said:
A jar contains red, blue, orange and green jelly beans. There are 20 of each color. How many jelly beens must be selected to guarantee that at least 3 are the same color?
The teacher got 9 as answer, but i do not know how she got this. Anybody would like to help please?

I think the question is 2 of each color ?
 
It is possible that after 8 draws, we have 2 of each of the four colors. Since the next one drawn must be one of the four colors, we will then have 3 of some color.

If, having drawn 8, we only have at most 3 colors of jelly beans, we must already have 3 of a matching color. For 3 pairs of 3 different colors is only 6 jelly beans, and we have two more, and any other combination of 6 jelly beans with 3 colors already includes 3 of one color.

It is also possible that after 8 draws, we have all four colors, but only one of some color. In this case, we have 7 jelly beans with 3 colors, and as we saw above, we can make at most 3 pairs from these, the 7th must match one of the 3 colors.

One can continue this analysis with just two colors: it can be seen that we need a mere 5 jelly beans to ensure 3 of a kind, and with only one color, we need but 3.

So no matter what, the 9th jelly bean will make (at least) the 3rd bean of some color.
 
Assuming that we have 4 holes and each one is one of the mentioned colors. If we select a certain beans we place it at its corresponding color. The worst case scenario is having each hole filled with two beans after the 8th draw. The 9th draw will have to be the third bean at any of the holes. Pigeonhole principle.
 
Yes, one can easily adapt that method to find the minimum draw that must have $k$ of any color, given we have $n$ colors, which is:

$n(k-1) + 1$.

This can be shown by double induction on $k$ and $n$, and in the case at hand (n = 4, k = 3) gives:

4(3 - 1) + 1 = 8 + 1 = 9.

(Assuming of course, we have enough objects to begin with).
 

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