Binomial Theorem — Formula Sheet

(x+y)^{n} = {}^{n}C_{0}x^{n}y^{0} + {}^{n}C_{1}x^{n-1}y^{1} + ... + {}^{n}C_{n-1}x^{1}y^{n-1} + {}^{n}C_{n}x^{0}y^{n} = sum_{r=0}^{n} {}^{n}C_{r}x^{n-

(1+x)^{n} = sum_{r=0}^{n} {}^{n}C_{r}x^{r}

(1-x)^{n} = sum_{r=0}^{n} {}^{n}C_{r}(-1)^{r}x^{r}

2^{n} = {}^{n}C_{0} + {}^{n}C_{1} + ... + {}^{n}C_{n}

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

r^{th} term from end = T_{n-r+2}

Middle term when n is even: ((n+2)/2)^{th} term

Middle terms when n is odd: ((n+1)/2)^{th} term and ((n+3)/2)^{th} term

The general term is T(r+1)=nCr x^(n-r)y^r.

The coefficients in the binomial theorem are written as nCr, where nCr = n! / [r!(n-r)!].

For every positive integer n, (x+y)^n = nC0 x^n y^0 + nC1 x^(n-1) y^1 + ... + nCn x^0 y^n = sum_{r=0}^{n} nCr x^(n-r) y^r.

The (r+1)th term of the expansion of (x+y)^n is T_(r+1) = nCr x^(n-r) y^r.