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\centerline{\bf Math 365 -- Wednesday 2/20/19 -- Section 6.1: Basic counting}


\begin{try}For each of the following, use some combination of the sum and product rules to find your answer. Give an un-simplified numerical answer (i.e.\ give your answer as, say, $5*4$ instead of $20$), and explain it, saying which rule(s) you're using when.
\begin{enumerate}[(a)]
\item A particular kind of shirt comes in two different cuts - male and female, each in three color choices and five sizes. How many different choices are made available?
\item On a ten-question true-or-false quiz, how many different ways can a student fill out the quiz if they definitely answer all of the questions? How many ways if they might leave questions blank? 
\item How many 3-letter words (these don't have to be real words, just strings of letters) are there?
\item How many 3-letter words are there that end in a vowel?
\item How many 3-letter words are there that have no repeated characters?
\item How many 3-letter words are there that have the property that if they start in a vowel then they don't end in a vowel? (You'll want to break this into disjoint cases).
\item How many 2-letter passwords are there that are made up of upper and/or lower case letters?
\item How many 2-letter passwords are there that are made up of upper and/or lower case letters, but where at least one of the letters is upper-case? (Again, you'll want to break this into disjoint cases).
\end{enumerate}
To check your answers: (a) 30; (b) 1024; 59,049; (c) 17,576; (d) 3380; (e) 15,600; (f) 16,926; (g) 2704; (h) 2028.
\end{try}



\begin{try}For each of the following, use some combination of the sum, product, inclusion-exclusion, and division rules to find your answer. Give an un-simplified numerical answer, and explain it, saying which rule(s) you're using when. (Note, there are 26 letters and 5 vowels.)
\begin{enumerate}[(a)]
\item How many strings of three letters are there that satisfy the following:
\begin{enumerate}[(i)]
\item that contain exactly one vowel?
\item that contain exactly 2 vowels?
\item that contain at least 1 vowel? 
\end{enumerate}
\item How many 3-card hands are there from a 52-card deck, which\dots
\begin{enumerate}[(i)]
\item have no other restrictions?
\item are all hearts?
\item are all the same suit?
\item form a straight (like 2,3,4 in possibly mixed suits. Note that Ace, 2, 3 and Queen, King, Ace are both straights.) 
\item are not all the same suit? (use two of your previous answers)
\end{enumerate}
\item How many ways are there to seat 6 people at a round table with 6 chairs, if you're only paying attention to who is sitting next to whom? 
\item How many positive integers (strictly) less than 100 are there that are divisible by 2 and/or 3? 
\end{enumerate}
To check your answers: (a) 6615; 1575; 8315; (b) 22,100; 286; 1144; 768; 20,956;  (c) 60; (d) 66.
\end{try}



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\begin{try}~
\begin{enumerate}[(a)]
\item How many positive integers less than 1000
\begin{enumerate}[(i)]
\item  are divisible by 7?
\item  are divisible by 7 but not by 11?
\item  are divisible by both 7 and 11?
\item  are divisible by either 7 or 11?
\item are divisible by exactly one of 7 and 11?
\item are divisible by neither 7 nor 11?
\item have distinct digits?
\item have distinct digits and are even?
\end{enumerate}
\item How many strings of three decimal digits
\begin{enumerate}[(i)]
\item do not contain the same digit three times?
\item begin with an odd digit?
\item have exactly two digits that are 4s?
\end{enumerate}
\item How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits?

\item Suppose that at some future time every telephone in the world is assigned a number that contains a country code that is 1 to 3 digits long, that is, of the form X, XX, or XXX, followed by a 10-digit telephone number of the form NXX-NXX-XXXX (as described in Example 8 in the book). How many different telephone numbers would be available worldwide under this numbering plan?
\item How many injective functions are there from $\{a,b,c\}$ to $\{1,2,3,4\}$? 
\item How many surjective functions are there from $\{a,b,c,d\}$ to $\{1,2,3\}$? \\
{[Hint: If $f: \{a,b,c,d\} \to \{1,2,3\}$ is surjective, then exactly one of $1, 2$, or $3$ has a preimage of size 2. First choose which of those three elements has the larger preimage, then pick it's preimage, and then assign the other two preimages.]}
\end{enumerate}
 \end{try}
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