In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

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Question: In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

Options:

Correct Answer: 240

Exam Year: 2021

Solution:

The coefficient of x^3 is C(5,3) * (4)^2 = 10 * 16 = 160.

In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

Step 1: Identify the expression to expand, which is (x + 4)^5.
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 = x, b = 4, and n = 5.
Step 5: We want the term where x is raised to the power of 3, which means we need k = 5 - 3 = 2.
Step 6: Calculate the binomial coefficient C(5, 2), which is the number of ways to choose 2 from 5. This is calculated as 5! / (2! * (5-2)!) = 10.
Step 7: Calculate 4 raised to the power of 2, which is 4^2 = 16.
Step 8: Multiply the binomial coefficient by 4^2 to find the coefficient of x^3: 10 * 16 = 160.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the Binomial Theorem, which involves combinations and powers.
Combinatorics – The study of counting, arrangements, and combinations, particularly using binomial coefficients in this context.