<p>The coefficient of <math>x^3y^4z^2</math> in the expansion of <math>(2x - 3y + 4z)^9</math> is</p>
Step-by-Step Solution
Key Concept: Use the multinomial theorem to find the general term in the expansion and match the powers of x, y, and z to extract the required coefficient.
<p><strong>Solution:</strong></p><p>The general term in the expansion of <math>(2x - 3y + 4z)^9</math> is:</p><p><math>\frac{9!}{a_1! a_2! a_3!} \times (2x)^{a_1}(-3y)^{a_2}(4z)^{a_3}</math></p><p><math>= \frac{9!}{a_1! a_2! a_3!} \times 2^{a_1}(-3)^{a_2} \times 4^{a_3} \times x^{a_1}y^{a_2}z^{a_3}</math></p><p>For the coefficient of <math>x^3y^4z^2</math>, we have <math>a_1 = 3, a_2 = 4, a_3 = 2</math></p><p><math>\text{Coefficient} = \frac{9!}{3! \times 4! \times 2!} \times 2^3 \times (-3)^4 \times 4^2</math></p><p><math>= 1260 \times 8 \times 81 \times 16 = 1260 \times 10368 = 13063680</math></p>
Correct Answer: b