First, let’s review some sticks and stones type questions we learned about in Section 3.5.
Then we will modify this and apply the principle of inclusion/exclusion from Section 3.3.
1.
Suppose we have 10 cookies to give away to three children, Albie, Bertie, and Charlie.
(a)
How many ways can we distribute the cookies with no restrictions?
(b)
How many ways can we distribute the cookies if each child must get at least two cookies?
Hint.
Give each kid the minimum number of cookies first. How many ways are there to distribute the remaining cookies?
(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)?
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.
(a)
How many ways can we distribute the cookies if Albie does get more than 3 cookies (so at least 4)?
How many ways can we distribute the cookies if Albie does not get more than 3 cookies?
(b)
How many ways can we distribute the cookies if Bertie does get more than 3 cookies?
(c)
How many ways can we distribute the cookies if both Albie and Bertie do get more than 3 cookies?
(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.
(e)
How many ways can we distribute the cookies if neither Albie nor Bertie gets more than 3 cookies?
