What is the value of the 3rd term in the expansion of (a + b)^6?

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Question: What is the value of the 3rd term in the expansion of (a + b)^6?

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Correct Answer: 20ab^4

Solution:

The 3rd term in the expansion of (a + b)^6 is given by C(6, 2) * a^4 * b^2 = 15 * a^4 * b^2.

What is the value of the 3rd term in the expansion of (a + b)^6?

What is the value of the 3rd term in the expansion of (a + b)^6?

Step 1: Identify the expression we are expanding, which is (a + b)^6.
Step 2: Understand that the expansion follows the Binomial Theorem, which states that (x + y)^n = Σ (C(n, k) * x^(n-k) * y^k) for k = 0 to n.
Step 3: In our case, x = a, y = b, and n = 6.
Step 4: We need to find the 3rd term in the expansion. The 3rd term corresponds to k = 2 (since we start counting from k = 0).
Step 5: Use the formula for the k-th term: T(k+1) = C(n, k) * a^(n-k) * b^k.
Step 6: Substitute n = 6 and k = 2 into the formula: T(3) = C(6, 2) * a^(6-2) * b^2.
Step 7: 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 8: Substitute C(6, 2) back into the term: T(3) = 15 * a^4 * b^2.
Step 9: Therefore, the value of the 3rd term in the expansion of (a + b)^6 is 15 * a^4 * b^2.

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