Two distinct concepts: the greatest binomial coefficient and the numerically greatest term.

Greatest binomial coefficient in (1+x)^n: This is purely about C(n,r), independent of x. The maximum C(n,r) occurs at r = n/2 (if n even, one maximum) or r = (n+/-1)/2 (if n odd, two equal maxima). Formally, C(n, floor(n/2)) = C(n, ceil(n/2)).

Numerically greatest term in (a+b)^n: This depends on the actual values of a and b. The method uses the ratio T_{r+1}/T_r = (n-r+1)/r * |b/a|.

Step 1: Set T_{r+1}/T_r >= 1 and solve for r: r <= (n+1)|b|/(|a|+|b|) = m. Step 2: If m is an integer, both T_m and T_{m+1} are greatest (equal). If m is not an integer, T_{floor(m)+1} is the unique greatest term.

Example: Greatest term of (2+3)^9 at x where b/a=3/2. m = 10*3/5 = 6. Since m is integer, T_6 and T_7 are both greatest.

Important distinction: "Greatest coefficient" means the largest C(n,r)*a^{n-r}*b^r in absolute value, not just C(n,r). For (1+x)^n with x=1, both concepts coincide. For other values of x, they differ.

JEE typically asks for either the greatest term or the rank (position) of the greatest term. The ratio method is the standard approach.