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I have seen the expression the identity product in somewhere and I try to express it again as \begin{align} % \prod _{k=1}^K\left(1-x_{k}\right)=\sum _{k=1}^K \frac{(-1)^k}{k!}\underbrace{\sum _{n_1=1}^K \ldots \sum _{n_k=1}^K}_{n_1\neq n_2\neq \ldots \neq ~ n_k} \prod _{t=1}^k x_{n_t} % \end{align} However, I am not sure if it is correct expression or not ? Could you give me a hint ?

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The good expression would be (don't forget $k=0$ !) $ \prod_{k=1}^K(1-x_k)=\sum_{k=0}^K(-1)^k\sum_{1\leq i_1<\ldots You can see it directly by developing the product and playing with permutations, or using the relation coefficients and roots of a polynomials using symmetric functions, namely plugging $z=1$ in $ \prod_{i=1}^K(z-x_i)=\sum_{k=0}^Kz^{K-k}(-1)^k\sum_{1\leq i_1<\ldots

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    Looks very spiffy to me, Student.2012-07-26