Problem 4 A Long-expected Party Bilbo Baggins of Bag End announced that he would shortly be celebrating his eleventy-first birthday with a party of special magnificence, there was much talk and excitement in Hobbiton. For subproblems 2, 3, and 4, reduce your answer to an integer and show your work. For subproblems 1 and 5, a concise mathematical expression suffices. i) Suppose Bilbo invited 111 guests. In how many ways might the guests arrive at the party if they arrive one at a time? ii) Suppose the guests include 11 Proudfoots (Proudfeet! ) consisting of 4 women, 4 men, and 3 children. In how many ways might the Proudfeet arrive if all the women enter together (one at a time), all the men enter together, and all the children enter together? (The three groups may still arrive in any order.) iii) Each table has a flower arrangement with 11 flowers. Bilbo makes the flower arrangement by using any combination of roses, daisies, irises, lilies, and elanor (a golden star-shaped flower that grows in Lothlórien). How many different flower arrangements are possible?
Discrete Mathematics
Problem 4 A Long-expected Party
Bilbo Baggins of Bag End announced that he would shortly be celebrating his eleventy-first birthday with a party of special magnificence, there was much talk and excitement in Hobbiton. For subproblems 2, 3, and 4, reduce your answer to an integer and show your work. For subproblems 1 and 5, a concise mathematical expression suffices.
i) Suppose Bilbo invited 111 guests. In how many ways might the guests arrive at the party if they arrive one at a time?
ii) Suppose the guests include 11 Proudfoots (Proudfeet! ) consisting of 4 women, 4 men, and 3 children. In how many ways might the Proudfeet arrive if all the women enter together (one at a time), all the men enter together, and all the children enter together? (The three groups may still arrive in any order.)
iii) Each table has a flower arrangement with 11 flowers. Bilbo makes the flower arrangement by using any combination of roses, daisies, irises, lilies, and elanor (a golden star-shaped flower that grows in Lothlórien). How many different flower arrangements are possible?
iv)Bilbo has 11 silver knives, 11 silver forks, and 11 silver spoons. The knives are all indistin- guishable from each other as are the forks and spoons. In how many ways can Lobelia steal 11 pieces of silverware while ensuring she takes at least 2 knives, 2 forks, and 2 spoons.
v) What if Frodo Baggins and Lobelia Sackville-Baggins are two of the eleven guests at one of the tables and they must NOT be seated next to each other. How many distinct and valid seating arrangements are there for this table? Rotating guests around the table does not count as a distinct seating arrangement.
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