In the expansion of (x + 1/x)^6, what is the coefficient of x^0?

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Question: In the expansion of (x + 1/x)^6, what is the coefficient of x^0?

Options:

Correct Answer: 20

Solution:

The coefficient of x^0 in (x + 1/x)^6 is given by 6C3 = 20.

In the expansion of (x + 1/x)^6, what is the coefficient of x^0?

In the expansion of (x + 1/x)^6, what is the coefficient of x^0?

Step 1: Understand that (x + 1/x)^6 means we are expanding the expression (x + 1/x) raised to the power of 6.
Step 2: Recognize that in the expansion, we will have terms that include different powers of x.
Step 3: We want to find the coefficient of x^0, which means we need the term where x cancels out completely.
Step 4: In the expansion, each term can be represented as (x^k) * (1/x)^(6-k) for k = 0 to 6.
Step 5: To find x^0, we need k such that k - (6 - k) = 0, which simplifies to 2k - 6 = 0.
Step 6: Solving 2k - 6 = 0 gives k = 3.
Step 7: The term corresponding to k = 3 is (x^3) * (1/x)^(3) = x^0.
Step 8: The coefficient of this term can be found using the binomial coefficient, which is 6C3.
Step 9: Calculate 6C3, which is equal to 6! / (3! * (6-3)!) = 20.
Step 10: Therefore, the coefficient of x^0 in the expansion is 20.

Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to expand expressions of the form (a + b)^n.
Finding Coefficients – It assesses the ability to find specific coefficients in the expansion, particularly the coefficient of a term where the exponent of x is zero.
Combinatorial Selection – The solution involves using combinations (nCr) to determine how many ways to select terms from the expansion.