Find the coefficient of x^4 in the expansion of (x + 1)^6.

Find the coefficient of x^4 in the expansion of (x + 1)^6.

Step 1: Understand that we need to find the coefficient of x^4 in the expression (x + 1)^6.
Step 2: 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 3: In our case, a = x, b = 1, and n = 6.
Step 4: We want the term where x is raised to the power of 4, which means we need to find the term where k = 2 (since 6 - k = 4).
Step 5: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to C(6, 2) because C(n, k) = C(n, n-k).
Step 6: C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15.
Step 7: The coefficient of x^4 is C(6, 4) * 1^2. Since 1^2 = 1, the coefficient remains 15.
Step 8: Therefore, the coefficient of x^4 in the expansion of (x + 1)^6 is 15.

Binomial Expansion – The expansion of expressions of the form (a + b)^n using the binomial theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion, particularly using the binomial coefficient C(n, k).