Exam 6: Sets and Counting

Four commuter trains and two express buses depart from city A to city B in the morning, and three commuter trains and two express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train? ​ __________ possible ways

Evaluate the expression. ​ ​

Four soups, five entrees, and four desserts are listed on the ''Special'' menu at the Neptune Restaurant. How many different selections consisting of one soup, one entree, and one dessert can a customer choose from this menu? ​

A member of book club wishes to purchase two books from selection of ten books recommended for a certain month. In how many ways can she choose them? ​

Let A and B be subsets of a universal set U and suppose , , , and . Compute . ​

​How many two-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions are allowed? __________ code words

Let and . Compute: ​ __________ ​ __________ ​ __________ ​ __________

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 45 questions, each having five possible answers. How many different responses are possible if every question is answered?

Evaluate the expression. ​

If , , , , , , and , find . ​

Student Painters, which specializes in painting the exterior of residential buildings, has seven people available to be organized into four-person and three-person teams. ​ In how many ways can the four-person team be formed?

Data released by the Department of Education regarding the rate (percentage) of ninth-grade students that don't graduate showed that out of 50 states, ​ 14 states had an increase in the dropout rate during the past 2 years. ​ 18 states had a dropout rate of at least 30% during the past 2 years. ​ 21 states had an increase in the dropout rate and/or a dropout rate of at least 30% during the past 2 years. ​ How many states had both a dropout rate of at least 30% and an increase in the dropout rate over the 2-year period? ​ __________ ​ How many states had a dropout rate that was less than 30% but that had increased over the 2-year period? ​ __________

Let U denotes the set of all students who applied for admission to fresman class at faber college for the upcoming academic year, and let Select the correct choice of venn daigram to describe the above sets.

Evaluate the expression. ​

In how many ways can a quality-control engineer select a sample of 2 transistors for testing from a batch of 30 transistors? ​ __________ ways

Determine whether the statement is true or false. ​ The number of four-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 179.

An exam consists of five true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? ​ __________ ​ In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered? ​ __________

Find the smallest possible set (that is, the set with the least number of elements) that contains the given sets as subsets. ​ ​

Results of a Department of Education survey of SAT test scores in showed that states had an average composite SAT score of at least 1000 during the past 3 years. 17 states had an increase of at least 10 points in the average composite SAT score during the past 3 years. 9 states had both an average composite SAT score of at least 1000 and an increase in the average composite SAT score of at least 10 points during the past 3 years. ​ How many of these states had composite scores less than 1000 and showed an increase of At least 10 points over 3-year period?

A list of poker hands ranked in order from the highest to the lowest is shown in the table, along with a description and example of each hand. Use the table to answer the problem. ​ ​ If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of two pairs? ​