For all natural numbers \(n\in\mathbb N\) and any two elements \(x,y\in R\) of a ring \((R,+,\cdot)\), there is a closed formula for the sum
\[\begin{array}{rcl}
\sum_{k=0}^n{n\choose k}x^{n-k}y^k&=&
{n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n\\
&=&(x+y)^n
\end{array}\]
Equivalently, \((x+y)^n\) can be expanded to the sum \(\sum_{k=0}^n{n\choose k}x^{n-k}y^k\). The symbol \({n \choose k}\) denotes the binomial coefficients.
Further Reading
[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983
[1112] Graham L. Ronald, Knuth E. Donald, Patashnik Oren: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition
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