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公式: $C_r^n = \frac{n!}{r!(n-r)!}$

公式: $(x+y)^n=C_0^nx^ny^0+C_1^nx^{n-1}y^1+C_2^nx^{n-2}y^2+...+C_n^nx^{0}y^n$

範例一:
\begin{aligned}
  (x+y)^2 &= C_0^2x^2y^0+C_1^2x^1y^1+C_2^2x^0+y^2\\
  &= \dfrac{2!}{0!(2-0)!}x^2+\dfrac{2!}{1!(2-1)!}x^1y^1+\dfrac{2!}{2!(2-2)!}y^2\\
  &= \dfrac{2*1}{1*(2*1)}x^2+\dfrac{2*1}{1*(1*1)}xy+\dfrac{2*1}{(2*1)*0}y^2\\
  &= x^2+2xy+y^2
\end{aligned}

範例二:
\begin{aligned}
(x+y)^3 &= C_0^3x^3y^0+C_1^3x^2y^1+C_2^3x^1y^2+C_3^3x^0y^3\\
&= \dfrac{3!}{0!(3-0)!}x^3+\dfrac{3!}{1!(3-1)!}x^2y^1+\dfrac{3!}{2!(3-2)!}x^1y^2+\dfrac{3!}{3!(3-3)!}y^3\\
&= \dfrac{3*2*1}{1*(3*2*1)}x^3+\dfrac{3*2*1}{1*(2*1)}x^2y^1+\dfrac{3*2*1}{2(1*1)}xy^2+\dfrac{3*2*1}{(3*2*1)*0}y^3\\
&= x^3+3x^2y+3xy^2+y^3
\end{aligned}

範例三：

\begin{aligned}
(2x+3y)^4 &= C_0^4(2x)^4(3y)^0+C_1^4(2x)^3(3y)^1+C_2^4(2x)^2(3y)^2+C_3^4(2x)^1(3y)^3+C_4^4(2x)^0(3y)^4\\
&= \dfrac{4!}{0!(4-0)!}(2x)^4+\dfrac{4!}{1!(4-1)!}(2x)^3(3y)^1+\dfrac{4!}{2!(4-2)!}(2x)^2(3y)^2+\dfrac{4!}{3!(4-3)!}(2x)^1(3y)^3+\dfrac{4!}{4!(4-4)!}(3y)^4\
&= \dfrac{4*3*2*1}{0(4*3*2*1)}(2x)^4+\dfrac{4*3*2*1}{1*(3*2*1)}(2x)^3(3y)^1+\dfrac{4*3*2*1}{2(2*1)}(2x)^2(3y)^2+\dfrac{4*3*2*1}{3*(1*1)}(2x)^1(3y)^3+\dfrac{4*3*2*1}{(4*3*2*1)*0}(3y)^4\\
&= 16x^4+48x^3y+72x^2y^2+48xy^3+9y^4
\end{aligned}

範例四：

\begin{aligned}
(x-y)^2 &= C_0^2x^2y^0+C_1^2x^1y^1+C_2^2x^0y^2\\
&= \dfrac{2!}{0!(2-0)!}x^2+\dfrac{2!}{1!(2-1)!}x^1y^1+\dfrac{2!}{2!(2-2)!}y^2\\
&= \dfrac{2*1}{1(2*1)}x^2+\dfrac{2*1}{1*(1*1)}xy+\dfrac{2*1}{(2*1)*0}y^2\\
&= x^2-2xy+y^2
\end{aligned}




$y=ax^2+bx+c$ 求最高或最低點

$\dfrac{dx}{dy}=D_y=y'=2ax+b=0$
$2ax=-b$
$x=-\dfrac{b}{2a}$

$y=2x^2+6x-8$ 求最高或最低點
一般解法: $判別式有解，然後公式解，
x=-\dfrac{2}{3}，y=-\dfrac{25}{2}$
微分解: $\dfrac{dx}{dy}=D_y=y'=4x+6=0，x=-\dfrac{2}{3}，帶回y=-\dfrac{25}{2}$