Combintatins question trying to understand example in the book

In summary, the conversation discusses a problem of forming teams with certain constraints in a group of 12 people. The solution involves dividing the set of all possible teams into two subsets and using combinations to determine the number of teams that contain both members of a pair or neither of them. The questioner clarifies the reasoning behind using 10 choose 3 and 10 choose 5 instead of 12 choose 5 and 12 choose 4.
  • #1
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Hello everyone,

I'm trying to understand this example in the book:
It says:

Suppose 2 members of the group of 12 insist on working as a pair--any team must cointain either both, or neither. How many five-person teams can be formed?

Solution: Call the 2 memebers of the group that insist on working as a pair A and B. Then any team formed must contain both A and B or neither A nor B. The set of all possible teams can be partitioned into 2 subsets as shown below.

Because a team that contians both A aand B contains exactly 3 other people from the remaining 10 is the group, there are as many such teams as there are subsets of three people that can be chosen from the remaning ten.

So the answer is 10 choose 3 = 120.

Because a team that contians neither A nor B contains exactly 5 people from the reminaing ten, there are as many such teams as there are subsets of 5 people that can be chosen from the remaning ten.

10 choose 5 = 252.

Because the set of teams that contain both A and B is disjoint from the set of teams that ccontain neither A nor B, by the addition rule:

# of teams containing both A and B or neither A nor B =
# of teams containing both A and B +
# of teams containing neither A nor B.

= 120 + 252 = 372.

What i don't understand is,
For the first part, 10 choose 3, if 2 of the team members must work together, why is it 10 choose 3? Is it becuase u assume you already have chosen 2 of the people, so now you only have to worry about choosing 3 more people to toss in that group?

And for part 2:
its 10 choose 5, is this because you havn't picked anyone yet to be in the group and no restrictions are set so you still have any 5 to choose in the group, so its 10 choose 5?


but if there are 12 people in the group, why isn't it 12 choose 5 and 12 choose 4? how did they get 10?

Because another example in the book, says the total number of teams of 5 is 12 choose 5. Not 10 chose 5
Thanks!
 
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  • #2
mr_coffee said:
What i don't understand is,
For the first part, 10 choose 3, if 2 of the team members must work together, why is it 10 choose 3? Is it becuase u assume you already have chosen 2 of the people, so now you only have to worry about choosing 3 more people to toss in that group?
Yes, you're right
And for part 2:
its 10 choose 5, is this because you havn't picked anyone yet to be in the group and no restrictions are set so you still have any 5 to choose in the group, so its 10 choose 5?


but if there are 12 people in the group, why isn't it 12 choose 5 and 12 choose 4? how did they get 10?

Because another example in the book, says the total number of teams of 5 is 12 choose 5. Not 10 chose 5
Thanks!
You are considering the case when neither of the members of the pair are working on the team. So the only members that can be on the team, in this case, are the other 10.
 
  • #3
ahh thanks so much!
 

What are combinations and how are they relevant in understanding the example in the book?

Combinations are a mathematical concept that refers to the number of ways a set of items can be selected without repetition. In the context of the book, combinations are used to determine the possible outcomes of a situation, such as choosing a committee from a pool of candidates.

What is the formula for calculating combinations?

The formula for calculating combinations is nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items being selected. This formula is used to determine the number of combinations possible in a given situation.

Why are combinations important in scientific research?

Combinations are important in scientific research because they allow us to analyze and understand the different possible outcomes of a situation. This can help us make predictions, draw conclusions, and make informed decisions based on the data.

Can you provide an example of how combinations are used in the book?

One example of how combinations are used in the book is when the author is discussing the possible outcomes of choosing a committee of 3 members from a group of 10 candidates. By using the formula for combinations, the author is able to determine that there are 120 possible combinations of committee members.

How can understanding combinations benefit us in our daily lives?

Understanding combinations can benefit us in our daily lives by helping us make decisions based on the possible outcomes of a situation. It can also aid in problem-solving and critical thinking skills, as well as in making predictions and analyzing data.

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