\input color %allows colored answers
\def\underline#1{\vbox{\hbox{#1}\vskip 1pt\hrule\vskip-1pt}}
\long\def\question#1#2{%question and answer
\par #1
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{\color{red}%
#2
}\vfil
}

{\bf \noindent\line{Math 3012-QHS \hfil Name: \rm \underline{Holden Rohrer}}
Fall 2019\par\noindent
Worksheet 1\par\noindent
Due 08/27/2019}
\smallskip \hrule \medskip
\question{
Q1 - The Greek alphabet consists of 24 letters. How many five-character strings can be made using the Greek alphabet (ignoring the distinction between uppercase and lowercase)?
}{
$24^5$ because the number of strings which can exist is the product of the number of the individual choices ($m = m_1\cdot m_2\ldots m_{n-1}\cdot m_n$), and each of the 5 individual choices has 24 options from the 24 letters.
}
\question{
Q2 - Assume that a license plate consists of 3 Latin alphabet letters followed by 4 numerals. How many license plates are there such that the numerals are distinct from one another and the last numeral is less than 3?

}{
By a similar method, the alphabet letters have $26^3$ possibilities. The numerals can be treated as choosing 2 distinct objects (for the last numeral), then ${9!}\over{6!}$ for the other 3, the options having been reduced by whichever chosen as the last numeral. This gives a final enumeration of $26^3\cdot 2 \cdot 504 = 17,716,608$.
}\eject

\question{
Q3 - Twenty-three students compete in a math competition in which the top three students are recognized with trophies for first, second, and third place. How many different outcomes are there for the top three places?
}{
$P(23,3)$ because order matters and there are three people chosen from twenty-three.
}

\question{
Q4 - Continuing the above problem---now assume that 3 additional students (distinct from the top 3) are awarded an honorable mention. How many different ways may non-placing students be awarded an honorable mention?
}{
Order doesn't matter, and there are now only 20 students to choose, from so there are ${20 \choose 3}$ ways to choose honorable mentions.
}\eject
\bye
