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

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

Step 1: Identify the expression to expand, which is (2x + 3)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * 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^2, which means we need to find the term where (2x) is raised to the power of 2.
Step 5: This corresponds to k = 4 because n - k = 2 (6 - k = 2). So, k = 4.
Step 6: Calculate the binomial coefficient 6C4, which is the same as 6C2 (since 6Ck = 6C(n-k)).
Step 7: 6C2 = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15.
Step 8: Calculate (2)^2, which is 4.
Step 9: Calculate (3)^4, which is 81.
Step 10: Multiply the results: 15 (from 6C2) * 4 (from (2)^2) * 81 (from (3)^4).
Step 11: Perform the multiplication: 15 * 4 = 60, then 60 * 81 = 4860.
Step 12: The coefficient of x^2 in the expansion of (2x + 3)^6 is 4860.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – The use of binomial coefficients (nCr) to determine the number of ways to choose terms from the expansion.
Exponentiation – Understanding how to correctly apply powers to the terms in the binomial expression.