What is the 4th term in the expansion of (2x - 3)^6? (2020)

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Question: What is the 4th term in the expansion of (2x - 3)^6? (2020)

Options:

Correct Answer: -540

Exam Year: 2020

Solution:

The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.

What is the 4th term in the expansion of (2x - 3)^6? (2020)

What is the 4th term in the expansion of (2x - 3)^6? (2020)

Step 1: Identify the expression to expand, which is (2x - 3)^6.
Step 2: 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 3: In our case, a = 2x, b = -3, and n = 6.
Step 4: We want the 4th term in the expansion. The 4th term corresponds to k = 3 (since we start counting from k = 0).
Step 5: Calculate C(6, 3), which is the number of combinations of 6 items taken 3 at a time. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 6: Calculate (2x)^(6-3) = (2x)^3 = 8x^3.
Step 7: Calculate (-3)^3 = -27.
Step 8: Combine these results to find the 4th term: C(6, 3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27.
Step 9: Multiply the numbers: 20 * 8 = 160, and then 160 * -27 = -4320.
Step 10: Therefore, the 4th term in the expansion of (2x - 3)^6 is -4320x^3.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find a specific term in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the coefficients of the terms in the expansion.
Exponentiation – Understanding how to apply exponents to both the coefficients and the variables in the binomial expression.