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Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.

In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?

Find the expansion of \((x+y)^{4}\) a) using combinatorial reasoning, as in Example \(1 .\) b) using the binomial theorem.

There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? b) In how many ways can one representative be picked who is either a mathematics major or a computer science major?

Find the expansion of \((x+y)^{5}\) a) using combinatorial reasoning, as in Example \(1 .\) b) using the binomial theorem.

Show that if there are 30 students in a class, then at least two have last names that begin with the same letter.

A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. a) How many socks must he take out to be sure that he has at least two socks of the same color? b) How many socks must he take out to be sure that he has at least two black socks?

How many permutations of {a, b, c, d, e, f, g} end with a?

A multiple-choice test contains 10 questions. There are four possible answers for each question. a) In how many ways can a student answer the questions on the test if the student answers every question? b) In how many ways can a student answer the questions on the test if the student can leave answers blank?

Every day a student randomly chooses a sandwich for lunch from a pile of wrapped sandwiches. If there are six kinds of sandwiches, how many different ways are there for the student to choose sandwiches for the seven days of a week if the order in which the sandwiches are chosen matters?