What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?

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Question: What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?

Options:

Correct Answer: 126

Solution:

The coefficient of x^5 is C(7,5) * (3)^2 = 21 * 9 = 189.

What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?

What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?

Step 1: Identify the expression we need to expand, which is (x + 3)^7.
Step 2: Recognize that we want the coefficient of x^5 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 = 3, and n = 7.
Step 5: To find the coefficient of x^5, we need to set n-k = 5, which means k = 2 (since 7 - 5 = 2).
Step 6: Calculate C(7, 2), which is the number of ways to choose 2 from 7. C(7, 2) = 7! / (2!(7-2)!) = 21.
Step 7: Now, calculate 3^2, which is 9.
Step 8: Multiply the coefficient C(7, 2) by 3^2: 21 * 9 = 189.
Step 9: Therefore, the coefficient of x^5 in the expansion of (x + 3)^7 is 189.

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