MegaMatma: Symbol i wzór Newtona.

Symbol i wzór Newtona.

Symbol Newtona $\ {n \choose k}=\frac{n!}{k!\left(n-k\right)!},\ \ \ n,\ k \in N\ i\ k\le n$

${n \choose 0}={n \choose n}=1$

${n \choose k}={n \choose {n-k}}$

${n \choose k}+{n \choose {k+1}}={{n+1} \choose {k+1}}$

Wzór Newtona

${\left(a+b\right)}^n={n \choose 0}\cdot a^n+{n \choose 1}\cdot {a^{n-1}}\cdot b+{n \choose 2}\cdot a^{n-2}\cdot b^2+\ \dots +{n \choose {n-1}}\cdot a\cdot {b}^{n-1}+{n \choose n}\cdot b^n$

Trójkąt Pascala

$1$

$1 \ \ 1$

$1 \ \ 2 \ \ 1$

$1 \ \ 3 \ \ 3 \ \ 1$

$1 \ \ 4 \ \ 6 \ \ 4 \ \ 1$

itd.

$\left( \begin{array}{c}0 \\0 \end{array}\right)$

$\left( \begin{array}{c}1 \\0 \end{array}\right) \ \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$

$\left( \begin{array}{c}2 \\0 \end{array}\right) \ \left( \begin{array}{c} 2 \\ 1 \end{array} \right) \ \left( \begin{array}{c}2 \\2 \end{array}\right)$

$\left( \begin{array}{c}3 \\0 \end{array}\right) \ \left( \begin{array}{c}3 \\1 \end{array} \right) \ \left( \begin{array}{c}3 \\ 2 \end{array}\right) \ \left( \begin{array}{c} 3 \\3 \end{array}\right)$

$\left( \begin{array}{c} 4 \\ 0 \end{array} \right) \ \left( \begin{array}{c} 4 \\ 1 \end{array} \right) \ \left( \begin{array}{c} 4 \\ 2 \end{array} \right) \ \left( \begin{array}{c} 4 \\ 3 \end{array} \right) \ \left( \begin{array}{c} 4 \\ 4 \end{array} \right)$
itd.

$(a+b)^0=\underline{1}$

$(a+b)^1=\underline{1}a+\underline{1}b$

$(a+b)^2=\underline{1}a^2+\underline{2}ab+\underline{1}b^2$

$(a+b)^3=\underline{1}a^3+\underline{3}a^2b+\underline{3}ab^2+\underline{1}b^3$

$(a+b)^4=\underline{1}a^4+\underline{4}a^3b+\underline{6}a^2b^2+\underline{4}ab^3+\underline{1}b^4$

itd.

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