Mathematics · Ch 6 — Binomial Theorem

Summary

The Binomial Theorem gives the expansion of $(x + y)^n$ for any positive integer $n$ :

(x + y)^n = \binom{n}{0} x^n y^0 + \binom{n}{1} x^{n-1} y^1 + \dots + \binom{n}{r} x^{n-r} y^r + \dots + \binom{n}{n} x^0 y^n

T_{r+1} = \binom{n}{r} x^{n-r} y^r

The middle term(s) depend on $n$ :
- If $n$ is even, there is one middle term: $T_{\frac{n}{2}+1}$ .
- If $n$ is odd, there are two middle terms: $T_{\frac{n+1}{2}}$ and $T_{\frac{n+3}{2}}$ .
The binomial coefficients $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$ are symmetric: $\binom{n}{r} = \binom{n}{n-r}$ .
The sum of all binomial coefficients in the expansion of $(1+x)^n$ is $2^n$ (put $x=1$ ). The sum of coefficients at even positions equals the sum at odd positions, each being $2^{n-1}$ . …