What is the coefficient of x^3 in the expansion of (x + 5)^8?

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Question: What is the coefficient of x^3 in the expansion of (x + 5)^8?

Options:

Correct Answer: 3360

Solution:

The coefficient of x^3 is given by 8C3 * (5)^5 = 56 * 3125 = 175000.

What is the coefficient of x^3 in the expansion of (x + 5)^8?

What is the coefficient of x^3 in the expansion of (x + 5)^8?

Step 1: Identify the expression to expand, which is (x + 5)^8.
Step 2: Understand that we want the coefficient of x^3 in this expansion.
Step 3: Use the binomial theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = x, b = 5, and n = 8.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 5 (since 8 - 3 = 5).
Step 6: Calculate the binomial coefficient 8C5, which is the same as 8C3 (because C(n, k) = C(n, n-k)).
Step 7: Calculate 8C3 using the formula: 8C3 = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8*7*6)/(3*2*1) = 56.
Step 8: Now calculate (5)^5, which is 5 * 5 * 5 * 5 * 5 = 3125.
Step 9: Multiply the coefficient 56 by (5)^5: 56 * 3125 = 175000.
Step 10: The coefficient of x^3 in the expansion of (x + 5)^8 is 175000.

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 binomial coefficients (nCr) to determine the number of ways to choose terms from the expansion.
Powers of a Constant – Understanding how to handle the constant term raised to a power in the binomial expansion.