Can someone explain this example of principle of inclusion-exclusion?

In summary: In this case, the intersection of four or five jugglers is not considered because it is possible for all five jugglers to have between three and seven balls, making the intersections 0.
  • #1
nowimpsbball
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Homework Statement


Suppose 30 identical juggling balls are distributed to 5 different jugglers.
A) how many ways can the balls be distributed so that each juggler receives at least three balls?
B)In how many ways can the balls be distributed so that each juggler receives between 3 and 7 balls?


The Attempt at a Solution



A) I understand perfectly, the ans is C(19,4)...I get that
B) The answer is C(19,4)-5C(14,4)+C(5,2)C(9,4)-C(5,3)C(4,4) = 121
From the book "the number of distributions can be represented by solutions of the linear equation x1+x2+x3+x4+x5=30. In this case, each xi satisfies 3<=xi<=7. We proceed with the help of the principle of inclusion-exclusion . First consider all distributions with xi>=3, C(19,4). If Pi is the property that xi>=8, we count only those distributions that satisfy none of the properties Pi. The Total count is (19,4)-5C(14,4)+C(5,2)C(9,4)-C(5,3)C(4,4) = 121."
I will bold the parts I do not understand. I was sick the day this was went over in class. One last question, why do you not consider the intersect of four and five "jugglers"...other words C(5,2) is the intersect of 2 jugglers, C(5,3) is the intersect of 3 jugglers, at least that is how I perceive it. They must be 0, but why would they be 0 in this case?

Thanks
 
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  • #2
!The answer is C(19,4)-5C(14,4)+C(5,2)C(9,4)-C(5,3)C(4,4) = 121. The number of distributions can be represented by solutions of the linear equation x1+x2+x3+x4+x5=30. This means that each xi satisfies 3<=xi<=7. To find the number of ways, the principle of inclusion-exclusion is used. This means that all distributions with xi>=3 are counted first, which is C(19,4). Then, any distributions that satisfy any of the properties Pi, where Pi is the property that xi>=8, are subtracted from the total count. This leads to the solution of (19,4)-5C(14,4)+C(5,2)C(9,4)-C(5,3)C(4,4) = 121. The reason why C(5,2), C(5,3) are included is because they represent the number of ways in which two jugglers or three jugglers respectively can have more than seven balls. This means that the number of distributions in which two jugglers have more than seven balls is C(5,2)C(9,4). Similarly, the number of distributions in which three jugglers have more than seven balls is C(5,3)C(4,4).
 

1. What is the principle of inclusion-exclusion?

The principle of inclusion-exclusion is a counting technique used to calculate the number of elements in a union of multiple sets. It states that the total number of elements in the union of two or more sets is equal to the sum of the number of elements in each set, minus the number of elements that are common to all sets.

2. Can you provide an example of the principle of inclusion-exclusion?

For example, let's say there are three sets: A, B, and C. Set A has 10 elements, set B has 8 elements, and set C has 6 elements. The number of elements in the union of these three sets would be (10+8+6) - (2+2+2) = 18. This is because there are 2 elements that are common to all three sets, and we need to subtract them once from the total sum.

3. How is the principle of inclusion-exclusion useful in mathematics?

The principle of inclusion-exclusion is useful in solving various counting problems, such as finding the number of ways to arrange objects or counting the number of possible outcomes in a probability experiment. It also helps in understanding the relationships between different sets.

4. Is the principle of inclusion-exclusion applicable to more than three sets?

Yes, the principle of inclusion-exclusion can be applied to any number of sets. The formula for calculating the number of elements in the union of multiple sets remains the same, i.e., the sum of the number of elements in each set minus the number of elements that are common to all sets.

5. Are there any limitations to the principle of inclusion-exclusion?

Yes, the principle of inclusion-exclusion assumes that the sets being considered are finite and that the elements within each set are distinct. It also does not take into account any overlap between more than two sets, which may lead to undercounting in some cases.

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