\magnification 1900
\parindent 0pt
\parskip 10pt
\pageno=1
\hsize 7.2truein
\hoffset -0.35truein
\voffset -0.3truein
\vsize 10truein
\def\u{\vskip -10pt}
\def\v{\vfill}
\def\s{\vskip -5pt}
\def\r{\vskip -4pt}
\def\Z{\cal Z}
\def\hb{\hfil \break}
\def\hr{\vskip 5pt \hrule }
\def\S5{\Sigma_{n=1}^5}

\underbar{6.2:  Combinations with repetitions.}

The number of integral solutions to $\S5 x_i = 20$ 

\rightline{where $-2 \leq x_i \leq 7$ $\forall i$}

= The number of integral solutions to $\S5 y_i = 30$ 

\rightline{where $0 \leq y_i \leq 9$ $\forall i$}

Pf:  Let $y_i = x_i + 2$



\hr

The number of 30-combinations of the 

\rightline{multiset $\{ 9 \cdot a_1, 9 \cdot a_1, 
9 \cdot a_2, 
9 \cdot a_3, 
9 \cdot a_4, 
9 \cdot a_5\} =$}

 \v
\eject

Let $S = $ the set of integral solutions to $y_1 + y_2 + y_3 + y_4 + y_5  = 30$ {where $0 \leq y_i $ $\forall i$}

\vskip 20pt

Then $|S| = $ the number of permutations of $\{30 \cdot 1, 4 \cdot +\}$ = 

\vskip 20pt

For $i = 1, 2, 3, 4, 5$, \hb
let $A_i =$ the set of integral solutions to $y_1 + y_2 + y_3 + y_4 + y_5  = 30$ 

\rightline{where $10 \leq y_i $ }

Ex:  $(10, 5, 5, 5, 5) \in A_1$,  $(0, 20, 7, 2, 1) \in A_2$,  

\rightline{$(0, 0, 10, 10, 10)  \in A_3 \cap A_4 \cap A_5$}

Then $\overline{\cup_{i=1}^5 A_i} = $ the set of  of integral solutions to $\S5 y_i = 30$ 

\rightline{where $0 \leq y_i \leq 9$ $\forall i$}

\end

6.3:  Suppose each person in a group of $n$ friends brings a gift to a party.  In how many ways can the $n$ gifts be distributed so that no person receives their own gift.

Let the set of friends = $\{p_1, ..., p_n\}$ where $p_i = $ person $i$.  Let the set of gifts = $\{g_1, ..., g_n\}$ where $g_j$ = the gift brought by person $i$.

Suppose $F: \{p_1, ..., p_n\}  \rightarrow \{g_1, ..., g_n\}$, $F(p_i) = g_j$ iff person $p_i$ receives give $g_j$, the gift brought by person $i$.  If no person receives their own gift. Then $F(p_i) \not= g_i$.

In simpler notation:  $F: \{1, ..., n\}  \rightarrow \{1, ..., n\}$, such that $F(i) \not = i$















\end