Find the coefficient of x^5 in the expansion of (3x + 2)^6.

Find the coefficient of x^5 in the expansion of (3x + 2)^6.

Step 1: Identify the expression we need to expand, which is (3x + 2)^6.
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 = 3x, b = 2, and n = 6.
Step 5: We need to find the term where the power of x is 5. This happens when we take (3x)^(5) and (2)^(1).
Step 6: The term we are interested in is C(6, 5) * (3x)^5 * (2)^1.
Step 7: Calculate C(6, 5), which is the number of ways to choose 5 items from 6. This equals 6.
Step 8: Calculate (3)^5, which is 3 * 3 * 3 * 3 * 3 = 243.
Step 9: Calculate (2)^1, which is simply 2.
Step 10: Now, multiply these values together: 6 * 243 * 2.
Step 11: First, multiply 6 * 243 to get 1458.
Step 12: Then, multiply 1458 * 2 to get 2916.
Step 13: Therefore, the coefficient of x^5 in the expansion of (3x + 2)^6 is 2916.

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 correct term in the expansion.
Coefficient Calculation – The calculation of coefficients involves multiplying the combination by the appropriate powers of the terms in the binomial expression.