In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?

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Question: In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?

Options:

Correct Answer: 540

Solution:

The coefficient is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = 720.

In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?

In the expansion of (2x - 3y)^5, what is the coefficient of x^3y^2?

Step 1: Identify the expression to expand, which is (2x - 3y)^5.
Step 2: Recognize that we need to find the coefficient of the term x^3y^2 in the expansion.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 4: In our case, a = 2x, b = -3y, and n = 5.
Step 5: We need the term where x has the power of 3 and y has the power of 2. This means we need k = 2 (since y is -3y).
Step 6: Calculate n - k, which is 5 - 2 = 3. This means we need the coefficient for x^3 and y^2.
Step 7: Use the binomial coefficient C(5, 2) to find the number of ways to choose 2 from 5, which is 10.
Step 8: Calculate (2)^3, which is 8, because we have 3 x's (2x) in the term.
Step 9: Calculate (-3)^2, which is 9, because we have 2 y's (-3y) in the term.
Step 10: Multiply the results: C(5, 2) * (2)^3 * (-3)^2 = 10 * 8 * 9.
Step 11: Calculate 10 * 8 = 80.
Step 12: Then calculate 80 * 9 = 720.
Step 13: The final answer is that the coefficient of x^3y^2 in the expansion is 720.

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