📂Lemmas

Binomial Theorem Proof

Theorem

$$(x+y)^{n} = \sum_{r=0}^{n} {_n C _r} x^{r} y^{n-r}$$ Here, ${_n C _r}$ is defined as the Binomial Coefficient. $${_n C _r} = \binom{n}{r} = {{ n! } \over { r ! (n-r)! }}$$

Description

It’s surprisingly useful right after you learn it in high school. Because of its versatility, it allows for the derivation of many formulas quickly and is widely used across various fields.

Proof

When expanding $(x+y)^{n}$, the coefficient of $x^{r} y^{n-r}$ is $$(x+y)^{n} = (x+y)(x+y)(x+y) \cdots (x+y)$$ equivalent to selecting $x$ of $(x+y)$ $n$ times and $y$ $n-r$ times respectively. Therefore, the combination $_n C _r$ becomes the coefficient of $x^{r} y^{n-r}$, $$(x+y)^{n} = \sum_{r=0}^{n} {_n C _r} x^{r} y^{n-r}$$

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