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

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

Options:

Correct Answer: -80

Solution:

The coefficient of x^3 in (2 - x)^5 is given by 5C3 * 2^2 * (-1)^3 = 10 * 4 * (-1) = -40.

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

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

Step 1: Identify the expression we are working with, which is (2 - x)^5.
Step 2: Recognize that we need to find the coefficient of x^3 in this expansion.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 4: In our case, a = 2, b = -x, and n = 5.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 3.
Step 6: Calculate nCk, which is 5C3. This is the number of ways to choose 3 items from 5, calculated as 5! / (3! * (5-3)!).
Step 7: Calculate 5C3 = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10.
Step 8: Now calculate 2^(n-k) = 2^(5-3) = 2^2 = 4.
Step 9: Next, calculate (-1)^k = (-1)^3 = -1.
Step 10: Combine these results to find the coefficient: 5C3 * 2^2 * (-1)^3 = 10 * 4 * (-1).
Step 11: Calculate the final result: 10 * 4 = 40, and then 40 * (-1) = -40.

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 binomial coefficients (nCr) to determine the number of ways to choose terms in the expansion.
Negative Exponents – Understanding how to handle negative terms in the binomial expansion.