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HW  p. 111: 1 - 4, 7, 16, 17, 18; p. 40:  19; and 

{\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?
\vfill
\vfill
\vfill

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

How many different DNA sequences of length 2? 

\vfill

How many different DNA sequences of length 3?  

\eject

{\bf 2.6 Subsets}

Suppose a set $A$ has four elements (i.e., the cardinality of $A = |A| = 4$) 

The number of subsets of $A$ is
\vfill

The number of nonempty subsets of $A$ is
\vfill

Suppose we know proteins $A, B, C, D$ affect a particular biological reaction.  How many different experiments can be performed in order to analyze the effects of these proteins on the biological reaction?

\vfill

A pizza parlor offers 4 different toppings (sausage, onions, chicken, walnuts).  How many different types of pizzas can one order?   
\vfill
Suppose a set $B$ has  $n$ elements (i.e., $|B| = n$). The number of subsets of $B$ is




\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 $|S| = |S_1| +  |S_2|$.




Suppose a symbol can be either a number between 0 and 9 or a letter.
How many are symbols there?

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

\vfill
\vfill


\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 $S$.

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

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


4 TA's need to be assigned to 4 different classes.  How many different possible assignments are there?

\vfill


4 classes need to be assigned a TA.  There are 10 TAs.  How many different possible assignments are there?

\vfill

Defn:  $n! = n(n-1)(n-2)... (2)(1)$,  ~$0! = 1$

Thm 3.2.1:  If $r > n$, then $P(n, r) = 0$. \hfil \break
If $r \leq n$, then $P(n, r) = {n! \over (n - r)!}$

$P(0, 0) = $ \hfil $P(n, 0) = $ \hfil $P(n, 1) = $ \hfil $P(n, n) = $ \hfil

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

An {\it $r$-combination} of $S$ is an $r$-element subset of $S$ (ORDER DOES NOT MATTER).

$C(n, r) =$ number of $r$-combinations of $S$ where $|S| = n$.  

How many different math teams consisting of 4 people can be formed if there are 10 students from which to choose?


\vfill

Thm:  $C(n, r) = \left( \matrix{ n \cr r} \right)$
$=  {n! \over r!(n - r)!} = {P(n, r) \over r!}$

Cor:  $C(n, r) = C(n, n-r)$

Cor:  $C(n, r) = C(n-1, r-1) + C(n-1, r) $

Cor:  Pascal's Triangle.

Cor:  $\Sigma_{i = 0}^n \left( \matrix{ n \cr i} \right) = 2^n$


\hrule

How many different proteins containing 10 amino acids can be formed if the protein contains
5 alanines(A), 3 leucines (L), and 2 serines (S)?  


\end

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