Find the coefficient of x^4 in the expansion of (x + 1)^8.

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Question: Find the coefficient of x^4 in the expansion of (x + 1)^8.

Options:

Correct Answer: 70

Solution:

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

Find the coefficient of x^4 in the expansion of (x + 1)^8.

Find the coefficient of x^4 in the expansion of (x + 1)^8.

Step 1: Understand that we want to find the coefficient of x^4 in the expression (x + 1)^8.
Step 2: Recognize that (x + 1)^8 can be expanded using the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = x, b = 1, and n = 8.
Step 4: We need to find the term where the power of x is 4. This means we want the term where k = 4 (because x^(8-k) = x^4 when k = 4).
Step 5: Use the Binomial Coefficient C(8, 4) to find the coefficient of the term with x^4.
Step 6: Calculate C(8, 4) using the formula C(n, k) = n! / (k! * (n-k)!). Here, C(8, 4) = 8! / (4! * 4!).
Step 7: Calculate 8! = 40320, 4! = 24, so C(8, 4) = 40320 / (24 * 24) = 40320 / 576 = 70.
Step 8: Conclude that the coefficient of x^4 in the expansion of (x + 1)^8 is 70.

Binomial Expansion – The question tests understanding of the binomial theorem, which states that (a + b)^n can be expanded using binomial coefficients.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements, which is crucial for finding specific terms in the expansion.