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In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?

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Question: In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?

Options:

  1. 108
  2. 72
  3. 36
  4. 144

Correct Answer: 108

Solution: 

The coefficient is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.

In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?

Practice Questions

Q1
In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?
  1. 108
  2. 72
  3. 36
  4. 144

Questions & Step-by-Step Solutions

In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?
Correct Answer: 216
  • Step 1: Identify the expression to expand, which is (2x + 3y)^4.
  • Step 2: Recognize that we need to find the coefficient of the term x^2y^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 = 4.
  • Step 5: We want the term where x has an exponent of 2 and y has an exponent of 2, which means we need k = 2 (since y is raised to the power of 2).
  • Step 6: Calculate n - k, which is 4 - 2 = 2. This means we need the coefficient for C(4, 2).
  • Step 7: Calculate C(4, 2), which is the number of ways to choose 2 items from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
  • Step 8: Calculate (2)^2, which is 4.
  • Step 9: Calculate (3)^2, which is 9.
  • Step 10: Multiply the results: Coefficient = C(4, 2) * (2)^2 * (3)^2 = 6 * 4 * 9.
  • Step 11: Perform the multiplication: 6 * 4 = 24, then 24 * 9 = 216.
  • Step 12: Conclude that the coefficient of x^2y^2 in the expansion of (2x + 3y)^4 is 216.
  • Binomial Expansion – The question tests the understanding of the binomial theorem, which provides a way to expand expressions of the form (a + b)^n.
  • Combination Formula – The use of the combination formula C(n, k) to determine the number of ways to choose k successes in n trials is essential for finding the coefficient.
  • Power of Terms – Understanding how to calculate the powers of the individual terms in the expansion is crucial for determining the specific coefficient.
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