Coefficient of $x^{11}$ in the expansion of $(1+x^{2})^{4}(1+x^{3})^{7}(1+x^{4})^{12}$ is

Coefficient of x¹¹ in the expansion of (1+x²)⁴(1+x³)⁷(1+x⁴)¹² is

$(1 + x^2)^4 (1 + x^3)^7 (1 + x^4)^{12}$

$\text{Find: coefficient of } x^{11}$

$(1+x^2)^4: x^{2a},\ 0\le a\le 4$

$(1+x^3)^7: x^{3b},\ 0\le b\le 7$

$(1+x^4)^{12}: x^{4c},\ 0\le c\le 12$

$2a + 3b + 4c = 11$

$c=0:\ 2a+3b=11 \Rightarrow (a,b)=(4,1),(1,3)$

$c=1:\ 2a+3b=7 \Rightarrow (a,b)=(2,1)$

$c=2:\ 2a+3b=3 \Rightarrow (a,b)=(0,1)$

$\text{coeff} = \sum \binom{4}{a}\binom{7}{b}\binom{12}{c}$

$(a,b,c)=(4,1,0):\ \binom44\binom71\binom{12}0 = 1\cdot7\cdot1 = 7$

$(1,3,0):\ \binom41\binom73\binom{12}0 = 4\cdot35\cdot1 = 140$

$(2,1,1):\ \binom42\binom71\binom{12}1 = 6\cdot7\cdot12 = 504$

$(0,1,2):\ \binom40\binom71\binom{12}2 = 1\cdot7\cdot66 = 462$

$7 + 140 + 504 + 462 = 1113$

$\boxed{\text{Coefficient of } x^{11} = 1113}$

$\text{Option (3)}$