Step 1: Formula for Numerically Greatest Term (N.G.T)
For expansion \( (A+B)^n \), find \( m = \frac{(n+1)|B|}{|A|+|B|} \).
Here \( A = 2x = 7 \), \( B = -3y = -3(3/7) = -9/7 \).
\( |A| = 7, |B| = 9/7 \).
Step 2: Calculate m
\[ m = \frac{(13+1) \cdot \frac{9}{7}}{7 + \frac{9}{7}} = \frac{14 \cdot \frac{9}{7}}{\frac{49+9}{7}} = \frac{18}{\frac{58}{7}} \times \frac{7}{7} \text{ (Wait, simplifying fraction)} \]
\[ m = \frac{14 \cdot (9/7)}{58/7} = \frac{14 \cdot 9}{58} = \frac{126}{58} \approx 2.17 \]
Since \( m \) is not an integer, the N.G.T is \( T_{[m]+1} \).
\( [m] = 2 \), so the greatest term is \( T_{2+1} = T_3 \).
Step 3: Calculate \( T_3 \)
\( T_{r+1} = \binom{n}{r} A^{n-r} B^r \). Here \( r=2 \).
\( T_3 = \binom{13}{2} (2x)^{11} (-3y)^2 \).
\( T_3 = \frac{13 \times 12}{2} (7)^{11} \left(\frac{9}{7}\right)^2 \).
\( T_3 = 78 \cdot 7^{11} \cdot \frac{3^4}{7^2} = 78 \cdot 7^9 \cdot 3^4 \).
Step 4: Match with Options
\( 78 = 2 \times 3 \times 13 = 26 \times 3 \).
\( T_3 = (26 \times 3) \cdot 3^4 \cdot 7^9 = 26 \cdot 3^5 \cdot 7^9 \).