In the expansion of (x + 5)^6, what is the coefficient of x^3?

₹0.0

Question: In the expansion of (x + 5)^6, what is the coefficient of x^3?

Options:

Correct Answer: 200

Solution:

The coefficient of x^3 is C(6, 3) * 5^3 = 20 * 125 = 250.

In the expansion of (x + 5)^6, what is the coefficient of x^3?

In the expansion of (x + 5)^6, what is the coefficient of x^3?

Step 1: Identify the expression to expand, which is (x + 5)^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 = x, b = 5, and n = 6.
Step 5: We want the term where x is raised to the power of 3, which means we need k = 3 (since n - k = 3).
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. This is calculated as 6! / (3! * (6-3)!) = 20.
Step 7: Calculate 5^3, which is 5 * 5 * 5 = 125.
Step 8: Multiply the coefficient C(6, 3) by 5^3 to find the coefficient of x^3: 20 * 125 = 250.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem, which involves combinations and powers.
Combinatorics – The use of combinations to determine the number of ways to choose elements from a set, specifically C(n, k) which represents the number of ways to choose k elements from n.
Powers of a Constant – Calculating the power of a constant (in this case, 5) raised to a specific exponent derived from the binomial expansion.