What is the coefficient of x^3 in the expansion of (x + 5)^6?

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Question: What is the coefficient of x^3 in the expansion of (x + 5)^6?

Options:

Correct Answer: 300

Solution:

The coefficient of x^3 in (x + 5)^6 is C(6, 3) * (5)^3 = 20 * 125 = 2500.

What is the coefficient of x^3 in the expansion of (x + 5)^6?

What is the coefficient of x^3 in the expansion of (x + 5)^6?

Step 1: Identify the expression we are working with, which is (x + 5)^6.
Step 2: Recognize that we want to find the coefficient of x^3 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 = 6.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 3 (since x^(n-k) = x^3).
Step 6: Calculate n - k, which is 6 - 3 = 3. This means we will use 5^3 in our calculation.
Step 7: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. This is calculated as 6! / (3! * (6-3)!) = 20.
Step 8: Calculate 5^3, which is 5 * 5 * 5 = 125.
Step 9: Multiply the results from Step 7 and Step 8: 20 * 125 = 2500.
Step 10: Conclude that the coefficient of x^3 in the expansion of (x + 5)^6 is 2500.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, represented as C(n, k), which is essential for determining the coefficient of a specific term in the expansion.