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Chapter 7: Q. 7.27 (page 354)

If 101items are distributed among 10 boxes, then at least one of the boxes must contain more than 10 items. Use the probabilistic method to prove this result.

Short Answer

Expert verified

Yes, there is a box containing >10items.

Step by step solution

01

Given information

There are 101items that are distributed in among 10boxes.

02

Explanation

Define random variables Xi,i=1,…,10that marks the number of items in box i. Observe that

EXi=10110

Now, suppose that all boxes have ≤10items in them. Then, we have that

101=∑iEXi≤∑i10=100

03

Final Answer

101=∑iEXi≤∑i10=100

Which is a contradiction. So, there has to be a box that has>10 items.

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