In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

₹0.0

Question: In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

Options:

Correct Answer: 540

Exam Year: 2022

Solution:

The coefficient of x^4 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.

In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

Step 1: Identify the expression to expand, which is (3x - 2)^6.
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 (nCk * 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 want the term where the power of x is 4, which means we need (3x) raised to the power of 4.
Step 6: This means we need to find the term where k = 2 (since 6 - k = 4).
Step 7: Calculate the binomial coefficient: 6C2 = 6! / (2!(6-2)!) = 15.
Step 8: Calculate (3)^4, which is 81.
Step 9: Calculate (-2)^2, which is 4.
Step 10: Multiply the results: 15 * 81 * 4 = 4860.
Step 11: Conclude that the coefficient of x^4 in the expansion of (3x - 2)^6 is 4860.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations (nCr) to determine how many ways to choose terms from the expansion.
Exponent Rules – The question requires knowledge of how to handle exponents when expanding binomials.