What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)

₹0.0

Question: What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)

Options:

Correct Answer: 180

Exam Year: 2023

Solution:

The coefficient of x^3 in (3x + 2)^5 is given by 5C3 * (3)^3 * (2)^2 = 10 * 27 * 4 = 1080.

What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)

What is the coefficient of x^3 in the expansion of (3x + 2)^5? (2023)

Step 1: Identify the expression we need to expand, which is (3x + 2)^5.
Step 2: Recognize that we want the coefficient of x^3 in this 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 = 5.
Step 5: We need to find the term where the power of x is 3. This happens when we take (3x)^(3) and (2)^(2).
Step 6: Calculate the number of ways to choose 3 terms from 5, which is given by 5C3 (the binomial coefficient).
Step 7: Calculate 5C3, which is equal to 10.
Step 8: Calculate (3)^3, which is 27.
Step 9: Calculate (2)^2, which is 4.
Step 10: Multiply these values together: 10 * 27 * 4.
Step 11: Perform the multiplication: 10 * 27 = 270, and then 270 * 4 = 1080.
Step 12: Conclude that the coefficient of x^3 in the expansion of (3x + 2)^5 is 1080.

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 combinations (nCr) to determine how many ways to choose terms from the expansion is essential for solving the problem.
Power of a Binomial – Understanding how to apply the powers of the terms in the binomial expression to find the desired coefficient.