Find the coefficient of x^5 in the expansion of (x + 1)^7.

Find the coefficient of x^5 in the expansion of (x + 1)^7.

Step 1: Understand that we need to find the coefficient of x^5 in the expression (x + 1)^7.
Step 2: Recognize that (x + 1)^7 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 = 7.
Step 4: We want the term where x is raised to the power of 5, which means we need to find the term where k = 2 (because 7 - k = 5).
Step 5: Calculate C(7, 2), which is the number of ways to choose 2 items from 7. This is calculated as C(7, 2) = 7! / (2!(7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.
Step 6: The coefficient of x^5 in the expansion is therefore 21.

Binomial Theorem – The Binomial Theorem provides a way to expand expressions of the form (a + b)^n, where the coefficients can be determined using combinations.
Combinations – Understanding how to calculate combinations (n choose k) is essential for finding coefficients in binomial expansions.