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What is the coefficient of x^1 in the expansion of (x - 1)^5?
What is the coefficient of x^1 in the expansion of (x - 1)^5?
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1 question
Q1
What is the coefficient of x^1 in the expansion of (x - 1)^5?
-5
5
10
15
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The coefficient of x^1 is given by 5C1 * (-1)^4 = 5 * 1 = 5.
Questions & Step-by-step Solutions
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Q
Q: What is the coefficient of x^1 in the expansion of (x - 1)^5?
Solution:
The coefficient of x^1 is given by 5C1 * (-1)^4 = 5 * 1 = 5.
Steps: 9
Show Steps
Step 1: Identify the expression we are working with, which is (x - 1)^5.
Step 2: Understand that we want to find the coefficient of x^1 in the expansion of this expression.
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 = -1, and n = 5.
Step 5: We need to find the term where the power of x is 1, which means we want k = 4 (since 5 - k = 1).
Step 6: Calculate the binomial coefficient for k = 4, which is 5C4. This is equal to 5.
Step 7: Calculate (-1)^4, which is 1.
Step 8: Multiply the results from Step 6 and Step 7: 5 * 1 = 5.
Step 9: Conclude that the coefficient of x^1 in the expansion of (x - 1)^5 is 5.
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