Find the coefficient of x^5 in the expansion of (x + 3)^8.

Find 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 to find 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 C(8, 5), which is the number of ways to choose 5 items from 8. This is equal to C(8, 3) = 8! / (3! * (8-3)!) = 56.
Step 7: Calculate 3^3, which is 3 * 3 * 3 = 27.
Step 8: Multiply the coefficient C(8, 5) by 3^3 to find the coefficient of x^5: 56 * 27 = 1512.

Binomial Expansion – The question tests the understanding of the binomial theorem, which allows for the expansion of expressions of the form (a + b)^n.
Combination Formula – The use of the combination formula C(n, k) to determine the number of ways to choose k successes in n trials is essential for finding the coefficient.
Power of a Constant – Understanding how to calculate the power of a constant (in this case, 3 raised to the power of 3) is necessary for finding the final coefficient.