What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

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Question: What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

Options:

Correct Answer: 300

Solution:

The coefficient of x^4 in (x + 5)^6 is given by 6C4 * 5^2 = 15 * 25 = 375.

What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

Step 1: Identify the expression we are working with, which is (x + 5)^6.
Step 2: We need to find the coefficient of x^4 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 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 (because 6 - 4 = 2).
Step 6: Calculate the binomial coefficient 6C2, which is the number of ways to choose 2 items from 6. This is calculated as 6! / (2! * (6-2)!) = 15.
Step 7: Now, calculate 5^2, which is 25.
Step 8: Multiply the coefficient (6C2) by 5^2: 15 * 25 = 375.
Step 9: Therefore, the coefficient of x^4 in the expansion of (x + 5)^6 is 375.

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