What is the 3rd term in the expansion of (x + 4)^6? (2020)

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Question: What is the 3rd term in the expansion of (x + 4)^6? (2020)

Options:

Correct Answer: 360x^4

Exam Year: 2020

Solution:

The 3rd term is C(6,2) * (4)^2 * (x)^4 = 15 * 16 * x^4 = 240x^4.

What is the 3rd term in the expansion of (x + 4)^6? (2020)

What is the 3rd term in the expansion of (x + 4)^6? (2020)

Step 1: Identify the expression to expand, which is (x + 4)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: Determine the values: a = x, b = 4, and n = 6.
Step 4: Find the 3rd term in the expansion. The 3rd term corresponds to k = 2 (since we start counting from k = 0).
Step 5: Calculate C(6, 2), which is the number of combinations of 6 items taken 2 at a time. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 6: Calculate (4)^2, which is 16.
Step 7: Calculate (x)^(6-2), which is (x)^4.
Step 8: Combine the results: 3rd term = C(6, 2) * (4)^2 * (x)^4 = 15 * 16 * x^4.
Step 9: Multiply 15 and 16 to get 240.
Step 10: Write the final answer: The 3rd term is 240x^4.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find a specific term in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the coefficients of the terms in the expansion.
Powers of Terms – Understanding how to correctly apply the powers of the terms in the binomial expression.