Calculate the coefficient of x^5 in the expansion of (x - 3)^7. (2021)

Calculate the coefficient of x^5 in the expansion of (x - 3)^7. (2021)

Step 1: Identify the expression we need to expand, which is (x - 3)^7.
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 = -3, and n = 7.
Step 4: We want to find the coefficient of x^5, which means we need to find the term where the power of x is 5.
Step 5: To find this term, we set n - k = 5, which means k = 7 - 5 = 2.
Step 6: Calculate C(7, 2), which is the number of ways to choose 2 items from 7. C(7, 2) = 7! / (2!(7-2)!) = 21.
Step 7: Now, we need to calculate (-3)^2, which is 9.
Step 8: Multiply the coefficient C(7, 2) by (-3)^2: 21 * 9 = 189.
Step 9: Since we are dealing with (-3), the final coefficient will be negative: -189.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose terms from the expansion.
Negative Coefficients – Understanding how to handle negative numbers in the expansion, particularly when raising a negative term to a power.