STAT 103.3Assignment 328 January 2022 1. Nine people are seated around a dinner table. Among them are two women with their (biological) children. Amanda has two girls, aged 7 and 9, and Beatrice has one boy who goes to kindergarten. The other diners are (adult) men, of which exactly two are professors. (a) For each of the following attributes, determine the number of individuals who satisfy the attribute, if possible. You may use common sense in your judgments, but avoid making inad- vertent assumptions beyond the facts stated. If there is insufficient information to determine the answer to the question, then say so, and explain why the answer is ambiguous. What is the number of i. females, ii. children under the age of 10, iii. professors, iv. sons who are not professors, v. people seated at the table with their parents? (b) The host suggests that a bridge game would be fun, but there is only room for four people. The other guests will entertain themselves by playing charades. It is decided to choose the bridge players at random, with all possible foursomes equally likely. (This protocol is called simple random sampling.) i. How many possible sets of bridge players can be selected from the nine guests, disregarding order? ii. Supposing that four people are selected, how many ways can they be arraged into two pairs of bridge partners? iii. What is the probability that none of the young children are chosen to play bridge? iv. What is the probability that all of the bridge players are male? 2. An assembly of nations wants to adopt new flags for each of its members. In the interest of uniformity, it is decided that each nation will have a flag consisting of three vertical bands of colour, with the colours being either red, green, blue, or white. [18] (a) How many nations can have different flags if there are no restrictions on more than one band being the same colour (so that one could could, for example, have a solid red flag created by three red bands)? (b) How many nations can have different flags if it is required that adjacent bands must be of different colours (so that e.g. red-white-red is allowed, but red-red-white is not)? (c) How many nations can have symmetrical flags which look the same from both sides? (e.g. red-white-red, but not red-white-green which would appear green-white-red from the other side.) (d) How many nations can have symmetrical flags if the restriction in ?? also applies. (e) How many nations can have different flags if the only restriction is that no nation may have a flag which is the reverse of another nation's flag. 1

3. Five couples seat themselves for dinner around a large circular table. (a) One of the diners, named Yvonne, observes the sequence of diners going counterclockwise from her right. How many possible sequences could there be, if there are no restrictions on the seating? (Consider the position of Yvonne as fixed, so that her spot would be the same under all arrangements.) (b) Suppose that after dinner everyone gets up to dance. After the dance, they all sit down at the table again in a random order. What is the probability that everyone again has the same neighbours? ("Having the same neighbours" includes the case where the same people may be on opposite sides, so that if Abigail had Zeke on her left and Yvonne on her right before the dance, but had Zeke on her right and Yvonne on her left after the dance, she still has the same neighbours.) (c) How many different arrangements are there such that everyone is sitting beside his or her partner? (d) Assuming that all partners sit next to each other before the dance, what is the probability that at least one couple are still next to each other after the dance?