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

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

Options:

Correct Answer: 45

Solution:

The coefficient of x^2 is given by 5C2 * (3)^3 = 10 * 27 = 270.

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

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

Step 1: Identify the expression we need to expand, which is (x + 3)^5.
Step 2: Understand that we want to find the coefficient of x^2 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 = x, b = 3, and n = 5.
Step 5: We need to find the term where x is raised to the power of 2, which means we need k = 3 (since 5 - 2 = 3).
Step 6: Calculate the binomial coefficient 5C3, which is the number of ways to choose 3 items from 5. This is calculated as 5! / (3! * (5-3)!) = 10.
Step 7: Calculate 3^3, which is 3 * 3 * 3 = 27.
Step 8: Multiply the binomial coefficient by 3^3 to find the coefficient of x^2: 10 * 27 = 270.

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