\magnification 2000
\parindent 0pt
\parskip 12pt
\pageno=1
\hsize 7.5truein
\hoffset -0.35truein
%%\voffset -0.1truein
\vsize 9.5truein
\def\u{\vskip -10pt}
\def\v{\vskip 5pt}
\def\s{\vskip -5pt}
\def\r{\vskip -4pt}

{\it Chapter 2 Basic Counting}

{\bf 2.1 Product Rule:}  If $S = S_1 \times S_2$, then $|S| = |S_1|  |S_2|$. 

$x = (a, b) \in S$ implies $a \in S_1$ AND $b \in S_2$, then $|S| = |S_1|  |S_2|$.




How many sequences consisting of one letter followed by one single digit number (0 - 9) are possible?



How many different license plates are possible if 3 letters followed by 3 numbers are used?


How many different DNA sequences of length 2?  

\eject

{2.6 Subsets}

\eject

{\bf 2.2 Sum Rule:}  If $S = S_1 \cup S_2$ and $S_1 \cap S_2 = \emptyset$, then $|S| = |S_1| +  |S_2|$. 

If $S_1 \cap S_2 = \emptyset$ and if $x \in S$ implies $x \in S_1$ OR $x \in S_2$, then $|S| = |S_1| +  |S_2|$.





How many symbols 

\hrule

Example 2:  How many 10-digit telephone numbers are there if 
\hfil \break
1.) The area code cannot begin with a 0 or 1 and must have a 0 or 1 in the middle. \hfil \break
2.) Neither of the first two digits of the last 7 digits can be 0 or 1.

\vfill

Example 3:  How many even numbers between 100 and 1000 have distinct digits.

\vfill
Example 4:  How many different seven-digit numbers can be constructed out 
of the digits 2, 4, 8, 8, 8, 8, 8?


\vfill

Example 5:  How many different seven-digit numbers can be constructed out
of the digits 2, 2, 8, 8, 8, 8, 8?


\eject

{\it 2.3, 2.5  Permutations and $r$-permuations:}

Suppose $|S| = n$.  

An {\it $r$-permutation} of $S$ is an ordered arrangement of $r$ of the $n$ elements of $S$.

\vfill

If $r = n$, then an $r$-permutation of $S$ is a  {\it permutation} of $S$.

\vfill

$P(n, r) =$ number of $r$-permutation of $S$ where $|S| = n$.  

If $r > n$, then $P(n, r) = $

$P(n, 1) = $

$n! = n(n-1)(n-2)... (2)(1)$

$0! = 1$

Thm 3.2.1:  If $r \leq n$, then $P(n, r) = {n! \over (n - r)!}$


\eject
 {\it 2.7  $r$-Combinations}

$\left( \matrix{ n \cr r} \right)$
$=  {n! \over r!(n - r)!}$

The number of $f$
\end

1, 4

$\emptyset$  $5^4$

(a) $5!$

(b)  (5^3)(2)

(a,b)  (4)(3)(2)2


\end