What is the value of the coefficient of x^5 in the expansion of (x + 3)^8?

What is the value of the coefficient of x^5 in the expansion of (x + 3)^8?

Step 1: Identify the expression we need to expand, which is (x + 3)^8.
Step 2: Understand 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 = 8.
Step 5: We need to find the term where x is raised to the power of 5, which means we need k = 3 (since 8 - 5 = 3).
Step 6: Calculate the binomial coefficient C(8, 3), which is the number of ways to choose 3 items from 8. This is calculated as 8! / (3! * (8-3)!) = 56.
Step 7: Calculate 3^3, which is 3 * 3 * 3 = 27.
Step 8: Multiply the coefficient C(8, 3) by 3^3 to find the coefficient of x^5: 56 * 27 = 1512.

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