Greatest binomial coefficient: In (1+x)^n, the largest C(n,r) occurs at r = n/2 (if n even) or r = (n-1)/2 and (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 = (n-r+1)/r * x. This ratio is > 1 when r < (n+1)x/(1+x). Let m = (n+1)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.