組合數有關公式求和

\[C_{n}^{m}=C_{n-1}^{m-1}+C_{n-1}^{m} \]

\[mC_{n}^{m}=nC_{n-1}^{m-1} \]

\[C_{n}^{0}+C _{n}^{1}+C_{n}^{2}+\ldots \ldots +C_{n}^{n}=2^n \]

\[1 C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+\ldots \ldots +nC_{n}^{n}=n2^{n-1} \]

\[1^{2}C_{n}^{1}+2^{2}C_{n}^{2}+3^{2}C_{n}^{3}+\ldots \ldots+n^{2}C_{n}^{n}=n(n+1)2^{n-2} \]

\[\dfrac{C_{n}^{1}}{1}-\dfrac{C_{n}^{2}}{2}+\dfrac{C_{n}^{3}}{3}+\ldots \ldots+\left( -1\right) ^{n-1}\dfrac{C_{n}^{n}}{n} = 1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots \ldots + \dfrac{1}{n} \]

轉載：https://blog.csdn.net/bigtiao097/article/details/77242624