# Exploring Your Chocolate Choices: 10 Combinations in 3 Selections

• MHB
• Hypatia1
In summary, the problem involves finding the maximum number of groups of 4 players that can be formed from a group of 16 players without any two players being in the same group more than once. This is similar to the social golfer problem, where a group of 32 golfers must be scheduled to play for as many weeks as possible without any two golfers playing in the same group more than once. The difference is that the latter problem also involves the additional variable of weeks.
Hypatia1
There are 10 different chocolates and you want to buy three of them. However, you cannot pick a pair of chocolate more than once. How many different choice you can make??

By "chocolates" do you mean 10 candy types or 10 individual candies?

Ten differrent type of chocolates
Lets say ten different brands

Hypatia said:
you want to buy three of them
And what about this phrase? Can I buy, say, 7 chocolates of 3 different types or does it have to be 3 individual chocolates?

Hypatia said:
you cannot pick a pair of chocolate more than once.
This is also unclear. If I pick a pair twice, I have 4 chocolates. Why is this a restriction if I need to buy just 3? Or, if I have to buy any number of chocolates of 3 different types, can I buy 3 chocolates of type 1? I have 1 pair and another single chocolate, so no pair is picked twice.

Please describe the problem statement as clearly as you can.

So let us say that we have the following chocolate brands
A B C D E
The number of three groups that can be formed with these brands will be 10. The order of your selection does not important and all three chocolate brands in each group should be different as the following
ABC
ABD
ABE
ACD
ACE
BCD
BCE
BDE
CDE
and let us say you buy A B C. From that point, but you can not buy chocolote groups involving A B, A C, or B C because you have chosen those pair of brands when you selected the ABC group. You cannot also buy A A A or A A B. Because that is not an option, as you see there will be 10 groups as listed above. When you select ABC, you can only buy ADE from the above list.

I'm not sure exactly what you are saying! It is true that, if we cannot have the same letter twice then we can choose any of the five letters first, then any of the remaining four, the any of the remaining 3 so 5(4)(3)= 60 different ways. But that includes the same three letters in different orders, such as ABC and BAC which apparently you do not want. There are 3!= 3(2)(1)= 6 different orders in which you can order three different ways. To discount those, divide by 3!= 6 to get 60/6= 10.

Selecting and buying mean the same. I used them interchangeably. Do you familiar with the social golfer problem? It is a very famous combinatorial optimization problem. Maybe I could not express as clearly as possible. But actually what I am asking is very similar to that of the social golfer problem. You can check it from the link below. https://en.m.wikipedia.org/wiki/Social_golfer_problem

But anyway, I want to write this problem here:

A group of 32 golfers plays golf once a week in groups of 4. Schedule these golfers to play for as many weeks as possible without any two golfers playing in the same group more than once.

But this problem, as you may notice, involves additional variable:week. Which I am not interested in. So let me phrase the problem again.

A group of 16 players plays golf once a week in groups of 4. Find the maximum number of groups for these golfers to play without any two golfers playing in the same group more than once.

How can we solve it?

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## 1. What is the purpose of "Exploring Your Chocolate Choices: 10 Combinations in 3 Selections"?

The purpose of this exploration is to provide a comprehensive guide to different chocolate combinations and selections, allowing individuals to make informed decisions when choosing their chocolate options.

## 2. How were the 10 combinations and 3 selections chosen for this exploration?

The 10 combinations and 3 selections were chosen based on extensive research and taste testing to ensure a diverse range of flavors and textures were represented.

## 3. Are there any health benefits to exploring different chocolate combinations?

While chocolate is generally considered a treat, certain combinations may offer health benefits. For example, dark chocolate paired with nuts or fruits can provide antioxidants and other nutrients.

## 4. Can these chocolate combinations be used in baking or cooking?

Yes, these combinations can be used in baking or cooking to add unique flavors and textures to your dishes. However, it is important to consider the overall balance of flavors and not overpower the other ingredients.

## 5. How can individuals use this exploration to expand their palate and try new chocolate combinations?

Individuals can use this exploration as a guide to try new chocolate combinations and selections, either by purchasing pre-made options or by experimenting with their own combinations at home. This can help expand their palate and discover new flavor profiles.

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