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What is the coefficient of x^6 in the expansion of (x + 5)^8?
What is the coefficient of x^6 in the expansion of (x + 5)^8?
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Q1
What is the coefficient of x^6 in the expansion of (x + 5)^8?
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The coefficient of x^6 is C(8,6) * (5)^2 = 28 * 25 = 700.
Questions & Step-by-step Solutions
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Q: What is the coefficient of x^6 in the expansion of (x + 5)^8?
Solution:
The coefficient of x^6 is C(8,6) * (5)^2 = 28 * 25 = 700.
Steps: 9
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Step 1: Identify the expression we are expanding, which is (x + 5)^8.
Step 2: Recognize that we want the coefficient of x^6 in this expansion.
Step 3: 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 4: In our case, a = x, b = 5, and n = 8.
Step 5: We need to find the term where x is raised to the power of 6, which means we need k = 2 (since 8 - 6 = 2).
Step 6: Calculate C(8, 2), which is the number of ways to choose 2 from 8. This is calculated as 8! / (2!(8-2)!) = 28.
Step 7: Calculate 5^2, which is 25.
Step 8: Multiply the results from Step 6 and Step 7: 28 * 25 = 700.
Step 9: Conclude that the coefficient of x^6 in the expansion of (x + 5)^8 is 700.
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