What is the coefficient of x^2 in the expansion of (x + 4)^5?

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Question: What is the coefficient of x^2 in the expansion of (x + 4)^5?

Options:

Correct Answer: 80

Solution:

The coefficient of x^2 is C(5,2) * 4^3 = 10 * 64 = 640.

What is the coefficient of x^2 in the expansion of (x + 4)^5?

What is the coefficient of x^2 in the expansion of (x + 4)^5?

Step 1: Identify the expression we need to expand, which is (x + 4)^5.
Step 2: Understand that we want the coefficient of x^2 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 = 4, and n = 5.
Step 5: We need to find the term where x is raised to the power of 2, which means we need k = 3 (because 5 - 2 = 3).
Step 6: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 items from 5. This is equal to 5! / (2! * (5-2)!) = 10.
Step 7: Calculate 4^3, which is 4 * 4 * 4 = 64.
Step 8: Multiply the coefficient C(5, 2) by 4^3: 10 * 64 = 640.
Step 9: Conclude that the coefficient of x^2 in the expansion of (x + 4)^5 is 640.

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 k successes in n trials is essential for calculating the coefficient.
Powers of Numbers – Understanding how to calculate powers, such as 4^3, is necessary for finding the final coefficient.