Explanation

(b)
$(1 + x + x^{2} + x^{3})^{6} = (1 + x)^{6}\left( 1 + x^{2} \right)^{6}$
$= \left( \ ^{6}C_{0} + \ ^{6}C_{1}x + \ ^{6}C_{2}x^{2} + \ ^{6}C_{3}x^{3} + \ ^{6}C_{4}x^{4} + \ ^{6}C_{5}x^{5} + \ ^{6}C_{6}x^{6} \right) \times (\ ^{6}C_{0} + \ ^{6}C_{1}x^{2} + \ ^{6}C_{2}x^{4} +$
$\ ^{6}C_{3}x^{6} + \ ^{6}C_{4}x^{8} + \ ^{6}C_{5}x^{10} + \ ^{6}C_{6}x^{12})$
$\therefore$ Coefficient of $x^{14}$ in ${(1 + x + x^{2} + x}^{3})^{6}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ ^{6}C_{2} \bullet \ ^{6}C_{6} + \ ^{6}C_{4} \bullet \ ^{6}C_{5} + \ ^{6}C_{6} \bullet \ ^{6}C_{4}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 15 + 90 + 15 = 120$