Exam 2: Combinatorial Metods

A valet who is completely color-blind is rearranging the cars of a parking lot. He has 5 Honda Civics, 4 Ford Fusions, 3 Land Rover LR1s and 3 Mercedes E500s. If the cars of the same model are indistinguishable how many ways can the valet park these in 3 rows of 5?

A hostess wants to seat a party of 9 around a circular table. John and Jennifer cannot sit next to each other. How many ways can she arrange the table?

A soccer team is made of 14 players. The game requires a structure of 4 forwards, 3 midfielders, 5 defenders, and 2 goal protectors. If each player can be assigned to any of the positions, how many ways can the players be assigned to these positions?

Prove the following binomial relationship: $\left( \begin{array} { c } 2 n \\n\end{array} \right) = \sum _ { j = 0 } ^ { n } \left( \begin{array} { l } n \\j\end{array} \right) ^ { 2 }$ Hint: Consider an urn containing red balls and white balls. Consider the number of ways to select balls from this urn at one time.

Three fair twenty-sided die are rolled (these have faces labelled 1-20). What is the probability they each show a prime number after rolled?

For a certain setlist, a band wants to pick 3 songs from each of their 3 albums. If one of the albums contains 10 songs, one contains 11 songs, and one contains 12 songs, how many possible setlists are possible?

A certain band consists of 20 players all of whom can play any instrument. They want to divide the players into a trio (group of 3), three quartets (group of 4), and a quintet (group of 5). How many ways can they do this?

Suppose that Janet bets on 12 different horse races and gets two correct. If she loses 7 of her tickets, what is the probability the remaining 5 have at least one winning ticket?

You play a game against a computer, which guesses your favorite Marvel comic book character. You answer a series of 15 yes or no questions, and the computer guesses your character based on the answers. If the computer is always correct, and for each sequence of yes's and no's there is a character, how many possible distinct characters are there?

A deli has 15 turkey and 7 ham sandwiches wrapped in aluminum foil unmarked. If they select 5 sandwiches at random, what is the probability they are all turkey?

15 students in Mrs. Studebacher's class will be arranged randomly into three rows of 5 for a class picture. What is the probability the tallest 5 students occupy the top row? (Assume all the students are distinct heights.)

An urn contains balls numbered one through 12. A person selects five balls at random and without replacement. What is the probability that they form a consecutive sequence of balls?

A company makes 50 deposits to their bank in one month. Four of the deposits are mistakenly written in the company's ledger as $2 more than actually deposited and three of the deposits are mistakenly recorded in the ledger as $2 less than was actually deposited. If the CEO checks the book keeping by comparing the sum of 6 random deposit receipts to the sum of their records in the ledger, what is the probability he thinks the ledger is accurate?

In a Bridge game, each of the 4 players is dealt 13 cards from a standard deck of cards at random. What is the probability two of the four players each get exactly two aces?

Gregory is new to a city and is speed dating. He can choose to go or not to go on a date with any number of 14 different suitors. If he goes on a date, he takes his date either to dinner or to see a movie, but not both. How many different options does Gregory have?