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{\bf Name: }
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{\bf Section: }

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\centerline {\bf Computer Science 243}
\centerline {\bf Spring 2025}
\centerline {\bf Homework 9}
% \centerline{\bf My name}
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\bf Due: beginning of class, Wednesday, 4/9/25 \hfill Points: 50
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Answer the following questions and show your work.  Your final submission must be completely your own work.

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\item {[5 points]}  Answer the following.  Show all work to receive full credit.
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\item {[3 points]} In a particular state, a driver's license identifier consists of either 2 uppercase letters followed by 3 digits [0-9], or 
3 uppercase letters followed by 2 digits. Furthermore, no driver's license identifier can start with a `X'. How many different 
driver's license identifiers are possible?
\item {[2 points]} How many strings of four lowercase English letters with at least one vowel can be created if 
(i) letters can be repeated, and (ii) if no letter can be repeated?
\end{enumerate}


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\item {[10 points]}  How many bit strings of length 8 
\begin{enumerate}[label=\alph*.]
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\item end in 01?
\item begin with 10 or end with 0?
\item contain exactly 3 0’s?
\item contain exactly 2 0’s or exactly 2 1's?
\item are palindromes?
\end{enumerate}


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\item {[9 points]}  Suppose that a password for a computer system must have at least 5, but no more than 7, characters, 
where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of three special 
characters \#, \$, or -.
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\item How many different passwords are possible for this system?
\item How many different passwords are possible if the password is required to contain exactly one of the special characters?
\item Using your answer in (a), how long would it take a hacker (in minutes) to try every possible password, assuming that it takes 1 ns to check a single password?
\end{enumerate}


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\item {[6 points]}  Thirteen people on a softball team show up for a game.
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\item How many ways are there to choose 9 players to take the field?
\item How many ways are there to assign the 9 positions by selecting players from the 13 people who show up?
\item Of the 13 people who show up, three are under 21.  How many ways are there to choose 9 players to take 
the field if at least one of these players must be under 21?
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\item {[6 points]}  Answer the following.  Show all work to receive full credit. 
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\item At least how many students must be present in class to guarantee that there are 30 either younger than 21 years old, 
or greater than or equal to 21 years old?
\item What is the minimum number of students, each of whom comes from one of the 50 U.S. states, who must 
be enrolled in a university to guarantee that there are at least 20 that come from the same state?
\item How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_1, b_1)$ and $(a_2, b_2)$ 
such that $a_1$ (mod 4) $= a_2$ (mod 4) and $b_1$ (mod 4) $= b_2$ (mod 4)?
\end{enumerate}


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\item {[4 points]}  Answer the following:
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\item Find the expansion of ${(3x + y)}^4$.  You may use Pascal's triangle, but take care to handle the coefficient of $x$.
\item What is the coefficient of $x^4y^{8}$ in the expansion of $(2x + y)^{12}$?
\end{enumerate}


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\item {[6 points]}  A bakery sells 6 different kinds of donuts: glazed, chocolate, cinnamon, custard, 
strawberry, and blueberry. Assuming order of selection does not matter, how many ways are there to select
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\item eight donuts?
\item a dozen donuts with at least two glazed and at least one blueberry?
\item ten donuts with no more than two chocolate?
\end{enumerate}


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\item {[4 points]}  Answer the following:
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\item How many different strings can be made using all of letters of MISSISSIPPI?
\item How many ways are there to distribute a deck of 52 cards to 4 players, with all players receiving 12 cards each?
\end{enumerate}


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