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A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. a) How many balls must she select to be sure of having at least three balls of the same color? b) How many balls must she select to be sure of having at least three blue balls?

Let S = {1, 2, 3, 4, 5}. a) List all the 3-permutations of S. b) List all the 3-combinations of S.

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

How many ways are there to assign three jobs to five employees if each employee can be given more than one job?

Undergraduate students at a college belong to one of four groups depending on the year in which they are expected to graduate. Each student must choose one of 21 different majors. How many students are needed to assure that there are two students expected to graduate in the same year who have the same major?

How many terms are there in the expansion of \((x+y)^{100}\) after like terms are collected?

What is the coefficient of \(x^{7}\) in \((1+x)^{11} ?\)

There are six professors teaching the introductory discrete mathematics class at a university. The same final exam is given by all six professors. If the lowest possible score on the final is 0 and the highest possible score is 100, how many students must there be to guarantee that there are two students with the same professor who earned the same final examination score?

Find the number of 5-permutations of a set with nine elements.

What is the coefficient of \(x^{9}\) in \((2-x)^{19} ?\)