Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)

Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)

Step 1: Identify the expression we need to expand, which is (2x - 3)^6.
Step 2: 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 3: In our case, a = 2x, b = -3, and n = 6.
Step 4: We want the coefficient of x^5, which means we need the term where (2x) is raised to the power of 5.
Step 5: This occurs when k = 1 because (2x)^(6-k) = (2x)^5 when k = 1.
Step 6: Calculate C(6, 5), which is the number of ways to choose 5 items from 6. This equals 6.
Step 7: Calculate (2)^5, which is 32.
Step 8: Calculate (-3)^1, which is -3.
Step 9: Multiply these values together: 6 (from C(6, 5)) * 32 (from (2)^5) * (-3) (from (-3)^1).
Step 10: The final calculation is 6 * 32 * (-3) = -576.

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 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.