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Discrete Mathematics An Open Introduction, 4th Edition

Oscar Levin

Worksheet Preview Activity
A4US

First, let’s review some sticks and stones type questions we learned about in Section 3.5.
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Then we will modify this and apply the principle of inclusion/exclusion from Section 3.3.
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1.
Suppose we have 10 cookies to give away to three children, Albie, Bertie, and Charlie.
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(a)
How many ways can we distribute the cookies with no restrictions?
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(b)
How many ways can we distribute the cookies if each child must get at least two cookies?
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Hint.
Give each kid the minimum number of cookies first. How many ways are there to distribute the remaining cookies?
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(c)
How many ways can you distribute the cookies if Albie gets at least 3 cookies and Bertie gets at least 2 cookies (and Charlie has no restrictions)?
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2.
Let’s again consider the 10 cookies we want to distribute to Albie, Bertie, and Charlie. This time, we will impose some upper bound restrictions.
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(a)
How many ways can we distribute the cookies if Albie does get more than 3 cookies (so at least 4)?
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How many ways can we distribute the cookies if Albie does not get more than 3 cookies?
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(b)
How many ways can we distribute the cookies if Bertie does get more than 3 cookies?
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(c)
How many ways can we distribute the cookies if both Albie and Bertie do get more than 3 cookies?
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(d)
Using the Principle of Inclusion/Exclusion for two sets, how many ways can we distribute the cookies if at least one of Albie or Bertie gets more than 3 cookies? So either Albie gets more than 3 cookies, Bertie gets more than 3 cookies, or both get more than 3 cookies.
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(e)
How many ways can we distribute the cookies if neither Albie nor Bertie gets more than 3 cookies?
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