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

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

Options:

Correct Answer: 54

Solution:

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

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

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

Step 1: Identify the expression to expand, which is (2 + 3x)^4.
Step 2: Recognize that we need to find the coefficient of x^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 = 2, b = 3x, and n = 4.
Step 5: We want the term where x has the power of 2, which means we need k = 2 (since (3x)^2 gives us x^2).
Step 6: Calculate C(4, 2), which is the number of ways to choose 2 from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 7: Calculate (2)^(4-2) = (2)^2 = 4.
Step 8: Calculate (3)^2 = 9.
Step 9: Multiply these values together: Coefficient = C(4, 2) * (2)^2 * (3)^2 = 6 * 4 * 9.
Step 10: Perform the multiplication: 6 * 4 = 24, then 24 * 9 = 216.
Step 11: Conclude that the coefficient of x^2 in the expansion of (2 + 3x)^4 is 216.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose terms from the expansion.
Exponent Rules – Understanding how to apply exponent rules when expanding the terms in the binomial expression.