Binomial expansion

The expansion (MM2.4)

• \(\binom nr = \dfrac{n!}{r!(n - r)!}\); the first few rows of Pascal's triangle (\(1\ 4\ 6\ 4\ 1\), \(1\ 5\ 10\ 10\ 5\ 1\), \(1\ 6\ 15\ 20\ 15\ 6\ 1\)) save time.
• \((1 + x)^n = 1 + nx + \binom n2 x^2 + \cdots\); \((a + bx)^n\): general term \(\binom nr a^{n-r}(bx)^r\).
• Symmetry \(\binom nr = \binom n{n-r}\); Pascal's rule \(\binom nr + \binom n{r+1} = \binom{n+1}{r+1}\); \(\sum_r \binom nr = 2^n\).
• Sum of coefficients: put \(x = 1\). Alternating sum: put \(x = -1\).

Techniques

• Term independent of \(x\) in \(\left(x^2 + \frac2x\right)^6\): the power of \(x\) is \(2(6 - r) - r = 12 - 3r\); set it to \(0\).
• Products such as \((1 + 3x)(1 - 2x)^4\): collect the two ways of making \(x^2\).
• Unknown \(n\) from two coefficients: divide the equations to eliminate the unknown constant, e.g. \(na = 12\), \(\binom n2 a^2 = 54\) gives \(\frac{n - 1}{2n} = \frac{54}{144}\).
• Approximations like \(1.01^5\): use the first few terms of \((1 + 0.01)^5\).