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\centerline{\bf Math 365 -- Monday 2/25/19}
\centerline{\bf Section 6.3 \& 6.2: Permutations, combinations, and pigeonhole principle}
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\begin{try}~
\begin{enumerate}[(a)]
\item Consider the set $\{a,b,c\}$. For each of the following, (A) list the objects described, (B) give a formula that tells you how many you should have listed, and (C) verify that the formula and the list agree.
\begin{enumerate}[(i)]
\item Permutations of $\{a,b,c\}$.
\item Two-permutations of $\{a,b,c\}$.
\item Size-two subsets of $\{a,b,c\}$.
\end{enumerate}
\item For each of the following, classify the problem as a permutation or a combination problem or neither, and give an answer using an unsimplified formula. (Answers should look, for example, like $5*4$ or $5!/2!$ instead of $P(5,4)$ or $20$.)
\begin{enumerate}[(i)]
\item In how many different orders can five runners finish a race if no ties are allowed?
\item How many strings of 1's and 0's of length seven have exactly three 1's?
\item How many strings of 1's and 0's of length seven have three or fewer 1's?
\item How many three-digit numbers are there with no 1's? (a three-digit number is something like 144 or 009 or 053)
\item How many three-digit numbers are there with no digits repeated?
\end{enumerate}
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\item For each of the following, provide your answers in an unsimplified form, and justify.
\begin{enumerate}[(i)]
\item A six-sided dice is rolled 5 times. How many ways could it turn out that a value greater than 4 (i.e.\ 5 or 6) is rolled exactly twice? \emph{(Hint: first pick which rolls are from $\{5,6\}$ (this implies which rolls are from $\{1,2,3,4\}$ for free), and then pick the values for the $\{5,6\}$-valued rolls, and finally pick the values for the other rolls (the $\{1,2,3,4\}$-valued rolls).)}
\item If 10 men and 10 women show up for one team of an intramural basketball game, how many ways can you pick 5 people to play for one team if there must be at least one person of each gender on the team?
\item How many ways are there for 5 women and 2 men to stand in line? Now how many ways are there for them to stand in line if the two men don't stand next to each other? (The men and the women are distinct individuals.)
\end{enumerate}
\item For each of the following identities, (A) explain in words why it makes sense given what it represents, and then (B) verify it algebraically using the formulas for permutation or combination.
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\emph{(For example, an answer for (A) might start out looking like ``$P(n,1)$ means\dots'', and an answer for part (B) should look like a calculation that starts with ``$P(n,1) = \dots$''.)}
\begin{enumerate}[(i)]
\item $P(n,1) = n$
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\item $P(n,0) = 1$
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\item $P(n,k+1) = P(n,k)*(n-k)$
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\item $\binom{n}{1} = n$
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\item $\binom{n}{n} = 1$
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\item $\binom{n}{0} = 1$
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\item $\binom{n}{k} = \binom{n}{n-k}$
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\end{enumerate}
\end{enumerate}
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\begin{try} For each of the following, be sure to include how the pigeonhole principle or its generalized version apply in your justifications, or why neither of them do.
\begin{enumerate}[(1)]
\item The lights have gone out and you're digging through an unorganized sock drawer filled with unmatched black socks and brown socks (otherwise roughly identical).
\begin{enumerate}
\item If you're pulling them out at random, how many socks do you need to take out to ensure you have a matching pair if there are 10 of each kind of sock? How about if there are 20 of each? 100 of each?
\item Again pulling at random, how many socks do you need to take out to ensure you have a matching brown pair if there are 10 of each kind of sock? How about if there are 20 of each? 100 of each?
\end{enumerate}
\item Explain why, out of any set of four integers, at least two have the same remainder when divided by 3.
\item A recent estimate showed that the US and Canada together (which share the country code $+$1) have approximately 134,000,000 phone lines in use. What is the minimum number of area codes needed to make that possible?
\item Let $f: A \to B$ be a function between finite sets such that $|A| > |B|$. Explain why $f$ cannot possibly be injective. (Consider the sizes of the preimages $\{f^{-1}(b) ~|~ b \in B\}$.)
\item Explain why, in any sequence of $n$ consecutive integers, at least one of them must be divisible by $n$. (Start with, say, $n=4$ as an example.)
\end{enumerate}
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