Find the coefficient of x^6 in the expansion of (x + 1)^10. (2023)

Find the coefficient of x^6 in the expansion of (x + 1)^10. (2023)

Step 1: Understand that we need to find the coefficient of x^6 in the expansion of (x + 1)^10.
Step 2: Recognize that the expansion can be done using 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 = 10.
Step 4: We want the term where x is raised to the power of 6, which means we need to find the term where k = 4 (since 10 - 6 = 4).
Step 5: Calculate the binomial coefficient for k = 4: This is 10C4.
Step 6: Use the formula for binomial coefficients: nCk = n! / (k! * (n-k)!). Here, n = 10 and k = 4.
Step 7: Calculate 10C4: 10! / (4! * (10-4)!) = 10! / (4! * 6!).
Step 8: Simplify the factorials: 10! = 10 × 9 × 8 × 7 × 6!, so we can cancel 6! in the numerator and denominator.
Step 9: Now we have: (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 10: Calculate the numerator: 10 × 9 = 90, 90 × 8 = 720, 720 × 7 = 5040.
Step 11: Calculate the denominator: 4 × 3 = 12, 12 × 2 = 24, 24 × 1 = 24.
Step 12: Now divide the numerator by the denominator: 5040 / 24 = 210.
Step 13: Therefore, the coefficient of x^6 in the expansion of (x + 1)^10 is 210.

Binomial Expansion – The question tests understanding of the binomial theorem, which states that (a + b)^n can be expanded using binomial coefficients.
Binomial Coefficients – The use of combinations (nCr) to find specific coefficients in the expansion.