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In the expansion of (x + 1/x)^8, what is the coefficient of x^4?

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Question: In the expansion of (x + 1/x)^8, what is the coefficient of x^4?

Options:

  1. 70
  2. 56
  3. 28
  4. 84

Correct Answer: 56

Solution: 

The coefficient of x^4 is C(8,4) = 70.

In the expansion of (x + 1/x)^8, what is the coefficient of x^4?

Practice Questions

Q1
In the expansion of (x + 1/x)^8, what is the coefficient of x^4?
  1. 70
  2. 56
  3. 28
  4. 84

Questions & Step-by-Step Solutions

In the expansion of (x + 1/x)^8, what is the coefficient of x^4?
Correct Answer: 70
  • Step 1: Identify the expression we are working with, which is (x + 1/x)^8.
  • Step 2: Recognize that we need to find the coefficient of x^4 in the 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 = 1/x, and n = 8.
  • Step 5: We want the term where the power of x is 4. This means we need to find k such that the exponent of x is 4.
  • Step 6: The exponent of x in the term C(8, k) * (x)^(8-k) * (1/x)^k is 8 - k - k = 8 - 2k.
  • Step 7: Set the equation 8 - 2k = 4 to find k. This simplifies to 2k = 4, so k = 2.
  • Step 8: Now, substitute k = 2 into the binomial coefficient C(8, k). This gives us C(8, 2).
  • Step 9: Calculate C(8, 2) = 8! / (2!(8-2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28.
  • Step 10: The coefficient of x^4 in the expansion of (x + 1/x)^8 is 28.
  • Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
  • Combinatorics – It involves calculating combinations, as the coefficient of a term in the expansion is determined by the binomial coefficient.
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