Find the coefficient of x^4 in the expansion of (3x - 2)^6.

Find the coefficient of x^4 in the expansion of (3x - 2)^6.

Step 1: Identify the expression we need to expand, which is (3x - 2)^6.
Step 2: Recall the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 3x, b = -2, and n = 6.
Step 4: We want the coefficient of x^4, which means we need to find the term where (3x) is raised to the power of 4.
Step 5: If (3x) is raised to the power of 4, then (-2) must be raised to the power of (6 - 4) = 2.
Step 6: The term we are looking for is given by the formula: nCk * (3x)^(n-k) * (-2)^k, where k = 2.
Step 7: Calculate nCk, which is 6C2. This is equal to 6! / (2!(6-2)!) = 15.
Step 8: Calculate (3)^4, which is 81.
Step 9: Calculate (-2)^2, which is 4.
Step 10: Multiply these values together: 15 * 81 * 4 = 4860.
Step 11: The coefficient of x^4 in the expansion of (3x - 2)^6 is 4860.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Coefficients in Expansion – Understanding how to find specific coefficients in the expansion of a binomial expression, particularly using the formula for combinations.