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

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

Options:

Correct Answer: 100

Solution:

The coefficient of x^2 is C(4,2) * (5)^2 = 6 * 25 = 150.

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

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

Step 1: Identify the expression we need to expand, which is (x + 5)^4.
Step 2: Understand that we want to find 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 = 5, and n = 4.
Step 5: We need to find the term where x is raised to the power of 2, which means we need k = 2 (because x^(n-k) = x^(4-2) = x^2).
Step 6: Calculate C(4, 2), which is the number of ways to choose 2 items from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 7: Calculate (5)^2, which is 25.
Step 8: Multiply the results from Step 6 and Step 7: 6 * 25 = 150.
Step 9: Conclude that the coefficient of x^2 in the expansion of (x + 5)^4 is 150.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations.