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Math 150 Exam 2
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November 3, 2006

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[10]~ 1a.) What is the coefficient of $x^3y^2z^5$ in the expansion of $(2x 
+ y - z)^{10}:~ \underline{  
(2^3)(-1)^5({10! \over 3!2!5!})
= {-8(10!) \over 3!2!5!}
}$

[6]~ 1b.) What is the coefficient of $x^3y^2z^4$ in the expansion of $(2x 
+ y - z)^{10}:~
\underline{0}$

[84]~ Choose 4 from the following 5 problems.  Circle your choices: A B C 
D E \break
 You may do all 5 problems in which case your  unchosen problem can
 replace your lowest problem at 4/5 the value.  
Note you must fully 
explain your answers.


A.)  Use Newtons binomial theorem to estimate $\sqrt{5}$ (expand to at 
least 4 terms).

$\sqrt{5} = (1 + 4)^{1\over 2} = 2({1 \over 4} + 1)^{1\over 2} = 2\Sigma_{k=0}^\infty 
\left(\matrix{{1 \over 2} \cr k}\right) ({1 \over 4})^k
\sim 2[1 + ({{1 \over 2} \over 1!}) ({1 \over 4}) + 
{ ({1 \over 2})({-1 \over 2}) \over 2!}  ({1 \over 4})^2 + 
{ ({1 \over 2})({-1 \over 2})({-3 \over 2}) \over 3!}  ({1 \over 4})^3]$ 
$ = 2[1 +  {1 \over 8} - {1 \over 128} + 
{1  \over 16}  ({1 \over 64})]$ 
$ = 2 + {1 \over 4} - {1 \over 64} + {1 \over 8} ({1 \over 64})$
$ = 2 + {1 \over 4} - {1 \over 64} + {1 \over 512}$



B.)  Find the number of integers between 1 and 10,000 inclusive that are not divisible by 4, 6, 10.

Similar to ch 6:  2


\end
C.)  What is the number of ways to place ten nonattacking rooks on the 10-by-10 board with forbidden positions as shown?

section 6.4

\includegraphics[width=24ex]{chess}

\vfill

D.)  Let $R_n$ denote the number of permutations of $X_n = \{1, 2, ..., n\}$, $n \geq 3$
in which neither the pattern 12 nor the pattern 23 occurs (note there are 
only 2 restrictions, for example, the pattern 34 may or may not occur).
Determine a formula for $R_n$ and prove your formula is correct.

section 6.4

E.)  Consider the partially ordered set $({\cal P}(X_2), \subset)$ of subsets of $\{1, 2\}$ partially ordered by containment.  Let a function $f$ in ${\cal F}({\cal P}(X_2))$ be defined by
$$f(A, B) = \cases{2 & if $A = B$ \cr
3 & if $A \subset B$, $A \not= B$ \cr
0 & otherwise}$$
\u
Find the following:

$f^{-1}(\emptyset, \emptyset) = \underline{2}$
\hfil
$f^{-1}(\emptyset, \{1\}) = \underline{3}$
\hfil
$f^{-1}(\emptyset, \{2\}) = \underline{3}$

$f^{-1}(\emptyset, \{1, 2\}) = \underline{3}$
\hfil
$(f*f)(\emptyset, \{1\}) = \underline{section 6.6}$
\hfil

\end