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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:  $\{ (2, 1),   (4, 3) \}$