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Find the coefficient of x^3 in the expansion of (x - 1)^7.
Find the coefficient of x^3 in the expansion of (x - 1)^7.
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Practice Questions
1 question
Q1
Find the coefficient of x^3 in the expansion of (x - 1)^7.
-35
-21
-7
-49
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The coefficient of x^3 is given by 7C3 * (-1)^4 = 35.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the coefficient of x^3 in the expansion of (x - 1)^7.
Solution:
The coefficient of x^3 is given by 7C3 * (-1)^4 = 35.
Steps: 9
Show Steps
Step 1: Identify the expression we need to expand, which is (x - 1)^7.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = x, b = -1, and n = 7.
Step 4: We want to find the term that contains x^3. This happens when n-k = 3, which means k = 7 - 3 = 4.
Step 5: Calculate the binomial coefficient for k = 4, which is 7C4.
Step 6: The formula for the binomial coefficient is nCk = n! / (k! * (n-k)!). So, 7C4 = 7! / (4! * 3!) = 35.
Step 7: The term we are interested in is 7C4 * (x)^(7-4) * (-1)^4.
Step 8: Substitute the values: 35 * x^3 * 1 = 35 * x^3.
Step 9: The coefficient of x^3 in the expansion is 35.
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