Greatest binomial coefficient: In (1+x)^n, the largest C(n,r) occurs at r = n/2 (if n even) or r = $\frac{n-1}{2}$ and $\frac{n+1}{2}$ (if n odd, both equal).

Greatest term (numerically largest): For (1+x)^n with x > 0, compute $T_{r+1}$/$T_r$ = $\frac{n-r+1}{r}$ * x. This ratio is > 1 when r < (n+1)$\frac{x}{1+x}$. Let m = (n+1)$\frac{x}{1+x}$. If m is integer, $T_m$ and $T_{m+1}$ are both greatest. If m is not integer, $T_{floor(m)+1}$ is the greatest term.