What is the coefficient of x^1 in the expansion of (x + 5)^6?

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Question: What is the coefficient of x^1 in the expansion of (x + 5)^6?

Options:

Correct Answer: 90

Solution:

The coefficient of x^1 is C(6,1) * 5^5 = 6 * 3125 = 18750.

What is the coefficient of x^1 in the expansion of (x + 5)^6?

What is the coefficient of x^1 in the expansion of (x + 5)^6?

Step 1: Identify the expression we are working with, which is (x + 5)^6.
Step 2: Understand that we want to find the coefficient of x^1 in the expansion of this expression.
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 the power of x is 1, which means we want k = 5 (since n - k = 1).
Step 6: Calculate C(6, 5), which is the number of ways to choose 5 items from 6. This is equal to 6.
Step 7: Calculate 5^5, which is 5 multiplied by itself 5 times. This equals 3125.
Step 8: Multiply the results from Step 6 and Step 7: 6 * 3125 = 18750.
Step 9: Conclude that the coefficient of x^1 in the expansion of (x + 5)^6 is 18750.

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