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

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

Options:

Correct Answer: 720

Solution:

Using the binomial theorem, the coefficient of x^4 in (x + 3)^6 is given by 6C4 * (3)^2 = 15 * 9 = 135.

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

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

Step 1: Identify the expression we are working with, which is (x + 3)^6.
Step 2: Recognize that we need to find the coefficient of x^4 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 = 6.
Step 5: We want the term where x is raised to the power of 4, which means we need k = 2 (since 6 - 4 = 2).
Step 6: Calculate the binomial coefficient 6C2, which is the number of ways to choose 2 items from 6. This is calculated as 6! / (2! * (6-2)!) = 15.
Step 7: Calculate 3^2, which is 9.
Step 8: Multiply the binomial coefficient by 3^2: 15 * 9 = 135.
Step 9: Conclude that the coefficient of x^4 in the expansion of (x + 3)^6 is 135.

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.
Combinations – Understanding how to calculate combinations (nCr) is essential for determining the coefficients in the binomial expansion.
Exponent Rules – Applying exponent rules correctly is necessary to identify the powers of x and the constant in the expansion.