In the expansion of (x - 2)^6, what is the coefficient of x^4?

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Question: In the expansion of (x - 2)^6, what is the coefficient of x^4?

Options:

Correct Answer: 90

Solution:

The coefficient of x^4 is given by C(6,4) * (-2)^2 = 15 * 4 = 60.

In the expansion of (x - 2)^6, what is the coefficient of x^4?

In the expansion of (x - 2)^6, what is the coefficient of x^4?

Step 1: Identify the expression to expand, which is (x - 2)^6.
Step 2: Recognize that we need to find the coefficient of x^4 in this 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 = x, b = -2, and n = 6.
Step 5: We want the term where x is raised to the power of 4, which means we need k = 2 (since 6 - k = 4).
Step 6: Calculate C(6, 2), which is the number of ways to choose 2 from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 7: Calculate (-2)^2, which is 4.
Step 8: Multiply the results from Step 6 and Step 7: 15 * 4 = 60.
Step 9: Conclude that the coefficient of x^4 in the expansion of (x - 2)^6 is 60.

Binomial Expansion – The question tests 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.
Negative Exponents – Understanding how to handle negative numbers in the binomial expansion.