Combinations possible when choosing 4 or 5 team members from

In summary, the question is asking for the number of combinations of 4 or 5 people from a group of 10. This can be solved using binomial coefficients, but when choosing 5, the answer must be divided by 2 due to the possibility of duplicate combinations. This can be avoided by considering an 11th "person" representing nobody, which would result in equivalent divisions of groups of 5. However, if the groups must be considered distinct, the division by 2 would not be necessary.
  • #1
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Homework Statement


How many combinations of people are there if you choose 4 or 5 from a group of 10?

Homework Equations


Relies on binomials

The Attempt at a Solution


binomial (10,4) = binomial (10,6) = 210
But when choosing 5 the answer is binomial (10,5) / 2 = 126
Why do I need to divide by 2?
 
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  • #2
I am not sure. Is the question to have 10 people, and sometimes you might choose 4 or maybe you choose 5? Then find the total number of combinations.
Could you just have an 11th "person" (person0) which means nobody, then choose 5 from 11? Sometimes one of the five "people" would be nobody, so you would just have 4 people.
 
  • #3
This depends on what divisions of "groups of 5" you consider equivalent. If it does not matter which group of 5 you end up in abcde|fghij is the same division as fghij|abcde. You do not have this issue in the case of splitting into one group of 4 and one of 6. However, if it does matter which group is which (e.g., all people in the first group gets a lollipop and the others do not) then those two divisions would be different and you would not divide by two.
 
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1. How many combinations are possible when choosing 4 team members from a group of 8?

There are 70 possible combinations when choosing 4 team members from a group of 8. This can be calculated using the combination formula, nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items being chosen.

2. What is the difference between choosing 4 team members and 5 team members from a group of 8?

The main difference is the number of combinations that are possible. When choosing 4 team members, there are 70 possible combinations. However, when choosing 5 team members, there are 56 more combinations, for a total of 126 possible combinations.

3. Can the order of the team members be changed in a combination?

No, the order of the team members does not matter in a combination. For example, choosing team members A, B, C, and D is the same combination as choosing D, C, B, and A. This is different from a permutation, where the order of the items does matter.

4. Is it possible for a team member to be chosen more than once in a combination?

No, in a combination, each team member can only be chosen once. This is because combinations are a way of selecting a subset of items from a larger group, without considering the order of the items.

5. How many combinations are possible when choosing 4 team members from a group of 10?

There are 210 possible combinations when choosing 4 team members from a group of 10. This can be calculated using the combination formula, nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items being chosen.

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