Question

The absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2x^2 + \frac{1}{2x}\right)^{11}\) is equal to:

• A. \(10^3 - 10\)
• B. \(12^3 - 12\)
• C. \(11^3 - 11\)
• D. \(13^3 - 13\)

Hint:

When dealing with expansions, simplify the general term first, then identify the required terms using the exponent relationships.

Answer 1

The Correct Option is B

To find the absolute difference of the coefficients of \(x^{10}\) and \(x^7\) in the expansion of \(\left(2x^2 + \frac{1}{2x}\right)^{11}\), we need to focus on the general term in the binomial expansion of \((a+b)^n\), which is given by:

\(T_k = \binom{n}{k} a^{n-k} b^k\)

For our problem, \(a = 2x^2\) and \(b = \frac{1}{2x}\), with \(n = 11\).

Thus, the general term \(T_k\) becomes:

\(T_k = \binom{11}{k} (2x^2)^{11-k} \left(\frac{1}{2x}\right)^k = \binom{11}{k} \cdot 2^{11-k} \cdot x^{2(11-k)} \cdot \frac{1}{2^k} \cdot x^{-k}\)

Simplifying the term further, we have:

\(T_k = \binom{11}{k} \cdot \frac{2^{11-k}}{2^k} \cdot x^{22-2k-k} = \binom{11}{k} \cdot 2^{11-2k} \cdot x^{22-3k}\)

To find the coefficient of \(x^{10}\), set the power of \(x\) to 10:

\(22 - 3k = 10\)

\(3k = 12\)

\(k = 4\)

So, the term is:

\(T_4 = \binom{11}{4} \cdot 2^{11-8} \cdot x^{10} = \binom{11}{4} \cdot 2^3\)

Calculate the coefficient:

\(\binom{11}{4} = \frac{11 \cdot 10 \cdot 9 \cdot 8}{4 \cdot 3 \cdot 2 \cdot 1} = 330\)

\(2^3 = 8\)

The coefficient of \(x^{10}\) is \(330 \cdot 8 = 2640\).

To find the coefficient of \(x^7\), set the power of \(x\) to 7:

\(22 - 3k = 7\)

\(3k = 15\)

\(k = 5\)

So, the term is:

\(T_5 = \binom{11}{5} \cdot 2^{11-10} \cdot x^7 = \binom{11}{5} \cdot 2^1\)

Calculate the coefficient:

\(\binom{11}{5} = \frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 462\)

The coefficient of \(x^7\) is \(462 \cdot 2 = 924\).

The absolute difference is given by:

\(|2640 - 924| = 1716\)

Calculate \(12^3 - 12\):

\(12^3 = 1728\)

\(12^3 - 12 = 1728 - 12 = 1716\)

Thus, the absolute difference matches the option \(12^3 - 12\), making it the correct answer.