MATH FACTORY

$ n $이 자연수일 때 $ (a+b)^n $의 전개식은 다음과 같다.

\begin{align*}
(a+b)^n &= \phantom{}_n\mathrm{C}_0 a^n + \phantom{}_n\mathrm{C}_1a^{n-1}b + \cdots + \phantom{}_n\mathrm{C}_ra^{n-r}b^r + \cdots + \phantom{}_n\mathrm{C}_nb^n \nonumber\\
&= \sum_{r=0}^{n}\phantom{}_n\mathrm{C}_ra^{n-r}b^r
\end{align*}

이를 이항정리라 하고, $ \phantom{}_n\mathrm{C}_ra^{n-r}b^r $을 일반항, $ \phantom{}_{n}\mathrm{C}_{0} $, $ \phantom{}_{n}\mathrm{C}_{1} $, $ \cdots $, $ \phantom{}_{n}\mathrm{C}_{n} $을 이항계수라 한다.

1. $ \phantom{}_n\mathrm{C}_0 + \phantom{}_n\mathrm{C}_1 + \phantom{}_n\mathrm{C}_2 + \phantom{}_n\mathrm{C}_3 + \cdots + \phantom{}_n\mathrm{C}_n = 2^n $
2. $ \phantom{}_n\mathrm{C}_0 - \phantom{}_n\mathrm{C}_1 + \phantom{}_n\mathrm{C}_2 - \phantom{}_n\mathrm{C}_3 + \cdots + (-1)^n\phantom{}_n\mathrm{C}_n = 0 $
3. $ \phantom{}_n\mathrm{C}_0 + \phantom{}_n\mathrm{C}_2 + \phantom{}_n\mathrm{C}_4 + \phantom{}_n\mathrm{C}_6 + \cdots = 2^{n-1} $
4. $ \phantom{}_n\mathrm{C}_1 + \phantom{}_n\mathrm{C}_3 + \phantom{}_n\mathrm{C}_5 + \phantom{}_n\mathrm{C}_7 + \cdots = 2^{n-1} $

1. $ \phantom{}_{n-1}\mathrm{C}_{r-1} + \phantom{}_{n-1}\mathrm{C}_{r} = \phantom{}_{n}\mathrm{C}_{r} $
2. $ \phantom{}_n\mathrm{C}_0 + \phantom{}_{n+1}\mathrm{C}_1 + \phantom{}_{n+2}\mathrm{C}_2 + \cdots + \phantom{}_{n+m}\mathrm{C}_m = \phantom{}_{n+m+1}\mathrm{C}_m $
3. $ \phantom{}_n\mathrm{C}_n + \phantom{}_{n+1}\mathrm{C}_n + \phantom{}_{n+2}\mathrm{C}_n + \cdots + \phantom{}_{n+m}\mathrm{C}_n = \phantom{}_{n+m+1}\mathrm{C}_{n+1} $

1. $ n=2k-1 $일 때, $ \phantom{}_n\mathrm{C}_0 + \phantom{}_n\mathrm{C}_1 + \cdots + \phantom{}_n\mathrm{C}_{k} = 2^{n-1} $
2. $ n=2k-1 $일 때, $ \phantom{}_n\mathrm{C}_{k+1} + \phantom{}_n\mathrm{C}_{k+2} + \cdots + \phantom{}_n\mathrm{C}_{n} = 2^{n-1} $
3. $ \displaystyle \sum_{r=0}^{n} \phantom{}_{n}\mathrm{C}_{r} \cdot \phantom{}_{n}\mathrm{C}_{n-r} = \phantom{}_{2n}\mathrm{C}_{n} $
4. $ (a+b+c)^n $의 전개식에서 $ a^p b^q c^r $의 계수는
   \begin{gather*}
   \frac{n!}{p!q!r!} \ \ (\textrm{단}, \ p+q+r=n)
   \end{gather*}