What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

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Question: What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

Options:

Correct Answer: 720

Exam Year: 2000

Solution:

The coefficient of x^3 is C(6,3) * (5)^3 * (2)^3 = 20 * 125 * 8 = 20000.

What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

What is the coefficient of x^3 in the expansion of (5x + 2)^6? (2000) 2000

Step 1: Identify the expression to expand, which is (5x + 2)^6.
Step 2: Recognize that we need to find the coefficient of x^3 in this expansion.
Step 3: 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 4: In our case, a = 5x, b = 2, and n = 6.
Step 5: We want the term where the power of x is 3, which means we need to find the term where (5x) is raised to the power of 3.
Step 6: This corresponds to k = 3 in the Binomial Theorem, since we want (5x)^(6-k) = (5x)^3.
Step 7: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 20.
Step 8: Calculate (5)^3, which is 125.
Step 9: Calculate (2)^(6-3) = (2)^3, which is 8.
Step 10: Multiply these values together: Coefficient = C(6, 3) * (5)^3 * (2)^3 = 20 * 125 * 8.
Step 11: Perform the multiplication: 20 * 125 = 2500, and then 2500 * 8 = 20000.
Step 12: Conclude that the coefficient of x^3 in the expansion of (5x + 2)^6 is 20000.

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.
Exponentiation – Understanding how to calculate powers of numbers, particularly in the context of the binomial expansion.