What is the coefficient of x^2 in the expansion of (x + 5)^7? (2021)

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Question: What is the coefficient of x^2 in the expansion of (x + 5)^7? (2021)

Options:

Correct Answer: 35

Exam Year: 2021

Solution:

The coefficient of x^2 is given by 7C2 * 5^5 = 21 * 3125 = 65625.

What is the coefficient of x^2 in the expansion of (x + 5)^7? (2021)

What is the coefficient of x^2 in the expansion of (x + 5)^7? (2021)

Step 1: Identify the expression to expand, which is (x + 5)^7.
Step 2: Understand that we need to find the coefficient of x^2 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 4: In our case, a = x, b = 5, and n = 7.
Step 5: We want the term where x is raised to the power of 2, which means we need k = 5 (because n - k = 2).
Step 6: Calculate the binomial coefficient 7C5, which is the same as 7C2 (because C(n, k) = C(n, n-k)).
Step 7: Calculate 7C2 = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.
Step 8: Now calculate 5^5, which is 5 * 5 * 5 * 5 * 5 = 3125.
Step 9: Multiply the coefficient 21 by 3125 to find the coefficient of x^2: 21 * 3125 = 65625.

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 (nCr) to determine how many ways to choose terms from the expansion is essential for solving the problem.
Powers of a Constant – Understanding how to calculate powers of constants (in this case, 5 raised to the power of 5) is necessary for finding the coefficient.