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When playing Yahtzee, you roll five regular 6-sided dice. How many different outcomes are possible from a single roll? The order of the dice does not matter.

A group of college students were asked about their TV watching habits. Of those surveyed, 28 students watch The Walking Dead, 19 watch The Blacklist, and 24 watch Game of Thrones. Additionally, 16 watch The Walking Dead and The Blacklist, 14 watch The Walking Dead and Game of Thrones, and 10 watch The Blacklist and Game of Thrones. There are 8 students who watch all three shows. How many students surveyed watched at least one of the shows?

Recall, by 8-bit strings, we mean strings of binary digits, of length 8 . (a) How many 8-bit strings are there total? (b) How many 8-bit strings have weight \(5 ?\) (c) How many subsets of the set \(\\{a, b, c, d, e, f, g, h\\}\) contain exactly 5 elements? (d) Explain why your answers to parts (b) and (c) are the same. Why are these questions equivalent?

An anagram of a word is just a rearrangement of its letters. How many different anagrams of "uncopyrightable" are there? (This happens to be the longest common English word without any repeated letters.)

How many permutations of \\{1,2,3,4,5\\} leave exactly 1 element fixed?

How many anagrams are there of the word "assesses" that start with the letter "a"?

What is the coefficient of \(x^{10}\) in the expansion of \((x+1)^{13}+x^{2}(x+1)^{17} ?\)

What is the coefficient of \(x^{12}\) in \((x+2)^{15} ?\)

How many integer solutions to \(x_{1}+x_{2}+x_{3}+x_{4}=25\) are there for which \(x_{1} \geq 1, x_{2} \geq 2, x_{3} \geq 3\) and \(x_{4} \geq 4 ?\)

What is the coefficient of \(x^{9}\) in the expansion of \((x+1)^{14}+x^{3}(x+2)^{15} ?\)