Category:Examples of Use of Binomial Theorem

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This category contains examples of use of the Binomial Theorem.

Let $X$ be one of the standard number systems $\N$, $\Z$, $\Q$, $\R$ or $\C$.

Let $x, y \in X$.

Then:

\(\ds \forall n \in \Z_{\ge 0}: \, \) \(\ds \paren {x + y}^n\) \(=\) \(\ds \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k\)
\(\ds \) \(=\) \(\ds x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots\)
\(\ds \) \(=\) \(\ds x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 3} } {3!} x^{n - 3} y^3 + \cdots\)

where $\dbinom n k$ is $n$ choose $k$.