\magnification 1800 \parindent 0pt \parskip 10pt \pageno=1 \hsize 7.2truein \hoffset -0.35truein \voffset -0.3truein \vsize 9.5truein \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 } 5.2: $(x + y)^n = \Sigma_{k = 0}^n \left(\matrix{n \cr k}\right) x^{k} y^{n-k}$ P.O.: The terms of $(x + y)^n$ are of the form $x^{k} y^{n-k}$. The coefficient of $x^{k} y^{n-k}$ = the number of ways to choose $k$ $x$'s and $(n-k)$ $y$'s = the number of ways to choose $k$ $x$'s from $n$ $x$'s = $\left(\matrix{n \cr k}\right)$. Alternatively, The coefficient of $x^{k} y^{n-k}$ = the number of ways to choose $k$ $x$'s and $(n-k)$ $y$'s = the number of permutations of the multiset \rightline{$\{k \cdot x, (n-k) \cdot y\} = \left(\matrix{n \cr k}\right)$.} \vfill \hr Obtain other formulas via substitution and algebraic manipulation such as differentiation. \hr Let $r \in {\cal R}$, $k \in {\cal Z}$. Define $\left(\matrix{r \cr k}\right) = \cases { {r(r-1) ... (r - k + 1) \over k!} & if $k \geq 1$ \cr 1 & if $k = 0$ \cr 0 & if $k \leq -1$ \cr } $ \eject Thm 5.3.1: Let $n$ be a positive integer. The sequence of binomial coefficients is a unimodal sequence. In particular if $n$ is even, \s\s\s\s\s\s\s $$ \left(\matrix{n \cr 0}\right) < \left(\matrix{n \cr 1}\right) ... < \left(\matrix{n \cr {n \over 2}}\right) $$ $$ \left(\matrix{n \cr {n \over 2}}\right) > ... > \left(\matrix{n \cr n-1}\right) > \left(\matrix{n \cr n}\right) $$ and if $n$ is odd \s\s\s $$ \left(\matrix{n \cr 0}\right) < \left(\matrix{n \cr 1}\right) ... < \left(\matrix{n \cr {n -1 \over 2}}\right) = \left(\matrix{n \cr {n +1 \over 2}}\right) $$ $$ \left(\matrix{n \cr {n + 1 \over 2}}\right) > ... > \left(\matrix{n \cr n-1}\right) > \left(\matrix{n \cr n}\right) $$ Proof idea: Look at ${ \left(\matrix{n \cr k }\right) \over \left(\matrix{n \cr k - 1 }\right) } = {n - k + 1 \over k} $ %%\eject \hr 5.4: Multinomial thm Define $\left(\matrix{n \cr n_1 n_2 ... n_t }\right) = {n! \over n_1! n_2! ... n_t!} $ Thm 5.5.1: Let $n \in {\cal Z}$. Then $$(x_1 + x_2 + ... x_t)^n = \Sigma \left(\matrix{n \cr n_1 n_2 ... n_t }\right) x_1^{n_1} x_2^{n_2} + ... x_t^{n_t}$$ where the summation extends over all nonnegative integral solutions to $n_1 + n_2 + ... + n_t = n$ \eject 5.5: Newton's Binomial Theorem Let $r \in {\cal R}$, $k \in {\cal Z}$. Define $\left(\matrix{r \cr k}\right) = \cases { {r(r-1) ... (r - k + 1) \over k!} & if $k \geq 1$ \cr 1 & if $k = 0$ \cr 0 & if $k \leq -1$ \cr } $ Thm 5.5.1: Let $\alpha \in {\cal R}$. Then if $ 0 \leq |x| < |y|$, $$(x + y)^\alpha = \Sigma_{k = 0}^\infty \left(\matrix{\alpha \cr k}\right) x^k y^{\alpha - k}$$ %%\vfill \end 4321: $\{ (4, 3), (4, 2), (4, 1), (3, 2), (3, 1), (2, 1) \}$ 4312: $\{ (4, 3), (4, 1), (4, 2), (3, 1), (3, 2) \}$ \hfil 3421: $\{ (4, 2), (4, 1), (3, 2), (3, 1), (2, 1) \}$ \hfil 4231: $\{ (2, 1), (3, 1), (4, 1), (4, 2), (4, 3) \}$ 4132: $\{ (4,1), (4, 2), (4, 3), (3,2) \}$ \hfil 3412: $\{ (3, 1), (3,2), (4, 1), (4, 2) \}$ \hfil 3241: $\{ (4, 1), (3, 2), (3, 1), (2, 1), \}$ \hfil 2431: $\{ (2, 1), (3, 1), (4, 1), (4, 3) \}$ \hfil 4213: $\{ (2, 1), (4, 1), (4, 2), (4, 3) \}$ 4123: $\{ (4,1), (4, 2), (4, 3) \}$ \hfil 1432: $\{ (4, 2), (4, 3), (3,2) \}$ \hfil 3142: $\{ (3, 1), (3,2), (4, 2) \}$ \hfil 3214: $\{ (3, 2), (3, 1), (2, 1) \}$ \hfil 2341: $\{ (2, 1), (3, 1), (4, 1) \}$ \hfil 2413: $\{ (2, 1), (4, 1), (4, 3) \}$ 1423: $\{ (4, 2), (4, 3) \}$ \hfill 1342: $\{ (4, 2), (3,2) \}$ \hfill 3124: $\{ (3, 1), (3,2) \}$ \hfill 2314: $\{ (2, 1), (3, 1) \}$ \hfill 2143: $\{ (2, 1), (4, 3) \}$