Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

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Question: Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

Options:

Correct Answer: 750

Solution:

The coefficient of x^4 is C(6,4) * 5^2 = 15 * 25 = 375.

Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

Calculate the coefficient of x^4 in the expansion of (x + 5)^6.

Step 1: Identify the expression to expand, which is (x + 5)^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 = 5, 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 - 4 = 2).
Step 6: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to C(6, 2) = 6! / (4! * 2!) = 15.
Step 7: Calculate 5^2, which is 25.
Step 8: Multiply the results from Step 6 and Step 7: 15 * 25 = 375.
Step 9: Conclude that the coefficient of x^4 in the expansion of (x + 5)^6 is 375.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.
Power of a Term – Identifying the specific term in the expansion that corresponds to a given power of x.