Question

The coefficient of \(xy^2z^3\) in the expansion of \((x - 2y + 3z)^6\) is:

• A. 6480
• B. 3240
• C. 1620
• D. 810

Hint:

When using the multinomial theorem, correctly identifying the powers and the coefficients of each variable is key to calculating the term in the expansion accurately.

Answer 1

The Correct Option is A

The coefficient of \(xy^2z^3\) in the polynomial expansion of \((x - 2y + 3z)^6\) is determined by identifying the term that matches the desired variables and their exponents. This is achieved through the multinomial expansion, where the specific term we are interested in is given by: \[ \binom{6}{1, 2, 3} \cdot 1^1 \cdot (-2)^2 \cdot 3^3 = \frac{6!}{1! \cdot 2! \cdot 3!} \cdot 1 \cdot 4 \cdot 27 = 60 \cdot 4 \cdot 27 = 6480 \] Thus, the coefficient is 6480.