Find the coefficient of x^3 in the expansion of (x + 1)^8.

Find the coefficient of x^3 in the expansion of (x + 1)^8.

Step 1: Understand that we need to find the coefficient of x^3 in the expression (x + 1)^8.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk) * a^(n-k) * b^k, where nCk is the binomial coefficient.
Step 3: In our case, a = x, b = 1, and n = 8.
Step 4: We want the term where x is raised to the power of 3, which means we need to find the term where k = 5 (since n - k = 3).
Step 5: Calculate the binomial coefficient 8C5, which is the same as 8C3 (because nCk = nC(n-k)).
Step 6: Use the formula for binomial coefficients: nCk = n! / (k! * (n-k)!). Here, n = 8 and k = 3.
Step 7: Calculate 8C3 = 8! / (3! * (8-3)!) = 8! / (3! * 5!).
Step 8: Simplify the factorials: 8! = 8 × 7 × 6 × 5!, so 8C3 = (8 × 7 × 6) / (3 × 2 × 1).
Step 9: Calculate the result: (8 × 7 × 6) = 336 and (3 × 2 × 1) = 6, so 336 / 6 = 56.
Step 10: Conclude that the coefficient of x^3 in the expansion of (x + 1)^8 is 56.

Binomial Expansion – The expansion of expressions of the form (a + b)^n using the binomial theorem, which involves combinations.
Combinatorics – The use of combinations to determine the coefficients of terms in the expansion.