the_binomial_theorem
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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| the_binomial_theorem [2020/11/03 20:47] – [계수] hkimscil | the_binomial_theorem [2020/11/18 00:20] (current) – hkimscil | ||
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| Line 78: | Line 78: | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| &{0 \choose 0}& | &{0 \choose 0}& | ||
| - | &{1 \choose 0}{1 \choose 1}& \\ | + | &{1 \choose 0} \; {1 \choose 1}& \\ |
| - | &{2 \choose 0}{2 \choose 1}{2 \choose 2}& \\ | + | &{2 \choose 0}\; {2 \choose 1}\; {2 \choose 2}& \\ |
| - | &{3 \choose 0}{3 \choose 1}{3 \choose 2}{3 \choose 3}& \\ | + | &{3 \choose 0}\; {3 \choose 1}\; {3 \choose 2}\; {3 \choose 3}& \\ |
| - | &{4 \choose 0}{4 \choose 1}{4 \choose 2}{4 \choose 3}{4 \choose 4}& \\ | + | &{4 \choose 0}\; {4 \choose 1}\; {4 \choose 2}\; {4 \choose 3}\; {4 \choose 4}& \\ |
| + | &{5 \choose 0}\; {5 \choose 1}\; {5 \choose 2}\; {5 \choose 3}\; {5 \choose 4}\; {5 \choose 5} & \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | & | + | & {\large |
| - | & 1 1 & \\ | + | & {\large |
| - | & 1 2 1 & \\ | + | & {\large |
| - | & 1 3 3 1 & \\ | + | & {\large |
| - | & 1 4 6 4 1 & \\ | + | & {\large |
| + | & {\large 1\quad 5\quad 10\quad 10\quad 5\quad | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| + | 따라서 | ||
| + | \begin{eqnarray*} | ||
| + | & a^5 + a^4 + a^3 + a^2 + a^1 + a^0 & \\ | ||
| + | & b^0 + b^1 + b^2 + b^3 + b^4 + b^5 & \\ | ||
| + | & 1 + 5 + 10 + 10 + 5 + 1 & \\ | ||
| + | & a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5 & \\ | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | |||
| 위를 종합해서 정리하면 | 위를 종합해서 정리하면 | ||
| Line 98: | Line 110: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | 예를 들면, 아래와 같다. ($n = 3$ 인경우) | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | (a + b)^{3} = & \sum^{3}_{k=0}{{3}\choose{k}} a^{3-k} b^{k} \\ | + | (a + b)^{3} |
| - | = & {{3}\choose{0}} a^{3-0} b^{0} + {{3}\choose{1}} a^{3-1} b^{1} + {{3}\choose{2}} a^{3-2} b^{2} + {{3}\choose{3}} a^{3-3} b^{3} \\ | + | & = & {{3}\choose{0}} a^{3-0} b^{0} + {{3}\choose{1}} a^{3-1} b^{1} + {{3}\choose{2}} a^{3-2} b^{2} + {{3}\choose{3}} a^{3-3} b^{3} \\ |
| - | = & 1 \cdot a^{3-0} b^{0} + 3 \cdot a^{3-1} b^{1} + 3 \cdot a^{3-2} b^{2} + 1 \cdot a^{3-3} b^{3} \\ | + | & = & 1 \cdot a^{3} b^{0} + 3 \cdot a^{2} b^{1} + 3 \cdot a^{1} b^{2} + 1 \cdot a^{0} b^{3} \\ |
| - | = & a^{3-0} b^{0} + 3a^{3-1} b^{1} + 3a^{3-2} b^{2} + a^{3-3} | + | & = & a^{3} + 3a^{2} b + 3ab^{2} + b^{3} |
| \end{eqnarray*} | \end{eqnarray*} | ||
the_binomial_theorem.1604404025.txt.gz · Last modified: 2020/11/03 20:47 by hkimscil