Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)

Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)

Step 1: Identify the expression to expand, which is (3x + 2)^5.
Step 2: Recognize that we need to find the coefficient of x^4 in the 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 = 5.
Step 5: We want the term where the power of x is 4, which means we need to choose (3x) four times and (2) one time.
Step 6: This corresponds to k = 1 (since we choose b = 2 one time).
Step 7: Calculate C(5, 1), which is the number of ways to choose 1 from 5. C(5, 1) = 5.
Step 8: Calculate (3)^4, which is 81.
Step 9: Calculate (2)^1, which is 2.
Step 10: Multiply these values together: 5 (from C(5, 1)) * 81 (from (3)^4) * 2 (from (2)^1).
Step 11: The final calculation is 5 * 81 * 2 = 810.

Binomial Expansion – The question tests the understanding of the binomial theorem, which is used to expand 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.
Coefficient Extraction – The ability to identify and calculate the specific coefficient of a term in the expanded form of a binomial expression.