Key Concept: General Term of a Binomial Expansion

The problem asks for the absolute difference of coefficients of specific terms in the expansion of a binomial expression. To find the coefficient of any particular power of $x$, we first need to determine the general term of the binomial expansion.

For a binomial expansion of the form $(a+b)^n$, the general term, often denoted as $T_{r+1}$, is given by the formula: $T_{r+1} = {}^{n}C_r \, a^{n-r} \, b^r$ where $n$ is the power of the binomial, and $r$ is the index of the term (starting from $r=0$ for the first term).

Step-by-Step Solution

1. Write Down the General Term of the Expansion

Our given expression is $\left(2 x^{2}+\frac{1}{2 x}\right)^{11}$. Here, $a = 2x^2$, $b = \frac{1}{2x}$, and $n = 11$.

Applying the general term formula: $T_{r+1} = {}^{11}C_r \, (2x^2)^{11-r} \, \left(\frac{1}{2x}\right)^r$

• Explanation: This step directly substitutes the components of our binomial into the general term formula. We need to expand this expression to separate the numerical coefficients and the powers of $x$.

2. Simplify the General Term

Now, we simplify the expression by distributing the exponents and combining like terms, especially powers of 2 and powers of $x$.

$\begin{aligned} T_{r+1} &= {}^{11}C_r \, 2^{11-r} \, (x^2)^{11-r} \, \left(\frac{1}{2}\right)^r \, \left(\frac{1}{x}\right)^r \\ &= {}^{11}C_r \, 2^{11-r} \, x^{2(11-r)} \, 2^{-r} \, x^{-r} \\ &= {}^{11}C_r \, 2^{11-r} \, x^{22-2r} \, 2^{-r} \, x^{-r} \\ &= {}^{11}C_r \, 2^{(11-r-r)} \, x^{(22-2r-r)} \\ &= {}^{11}C_r \, 2^{11-2r} \, x^{22-3r}\end{aligned}$

• Explanation:
  • We used the exponent rules $(a^m)^n = a^{mn}$ and $\left(\frac{1}{a}\right)^n = a^{-n}$.
  • All terms involving base 2 are grouped together, and all terms involving base $x$ are grouped together.
  • This simplified form, $T_{r+1} = {}^{11}C_r \, 2^{11-2r} \, x^{22-3r}$, is crucial because it clearly shows the coefficient and the power of $x$ for any given $r$.

3. Find the Coefficient of $x^{10}$

To find the coefficient of $x^{10}$, we set the power of $x$ in our simplified general term equal to 10:

$22 - 3r = 10$ $3r = 22 - 10$ $3r = 12$ $r = 4$

Now, substitute $r=4$ into the coefficient part of the general term (excluding $x^{22-3r}$):

Coefficient of $x^{10} = {}^{11}C_4 \, 2^{11-2(4)}$ $= {}^{11}C_4 \, 2^{11-8}$ $= {}^{11}C_4 \, 2^3$ $= {}^{11}C_4 \times 8$

• Explanation: By equating the exponent of $x$ from the general term to the desired exponent (10), we solve for the specific value of $r$ that yields that term. Once $r$ is found, we plug it back into the numerical and constant parts of the general term to get the full coefficient.

4. Find the Coefficient of $x^{7}$

Similarly, to find the coefficient of $x^{7}$, we set the power of $x$ in our simplified general term equal to 7:

$22 - 3r = 7$ $3r = 22 - 7$ $3r = 15$ $r = 5$

Now, substitute $r=5$ into the coefficient part of the general term:

Coefficient of $x^{7} = {}^{11}C_5 \, 2^{11-2(5)}$ $= {}^{11}C_5 \, 2^{11-10}$ $= {}^{11}C_5 \, 2^1$ $= {}^{11}C_5 \times 2$

• Explanation: The process is identical to finding the coefficient of $x^{10}$, just with a different target power for $x$.

5. Calculate the Absolute Difference of the Coefficients

We need to calculate the absolute difference between ${}^{11}C_4 \times 8$ and ${}^{11}C_5 \times 2$.

First, let's calculate the binomial coefficients:

${}^{11}C_4 = \frac{11!}{4!(11-4)!} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = 11 \times 10 \times 3 = 330$ ${}^{11}C_5 = \frac{11!}{5!(11-5)!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 11 \times 3 \times 2 \times 7 = 462$

Now, substitute these values back:

$\text{Coefficient of } x^{10} = 330 \times 8 = 2640$ $\text{Coefficient of } x^{7} = 462 \times 2 = 924$

The required absolute difference is: $|2640 - 924| = |1716| = 1716$

• Explanation: We compute the numerical values of the coefficients. The formula for combinations, ${}^nC_r = \frac{n!}{r!(n-r)!}$, is used. Simplifying the factorials by canceling common terms is an efficient way to calculate these values.

6. Match the Result with the Options

We have the difference as 1716. Let's evaluate the given options: (A) $11^3 - 11 = 1331 - 11 = 1320$ (This does not match)

Let's recheck the calculation in the original solution which led to $12^3 - 12$. The original solution shows: $\begin{aligned} & {}^{11} \mathrm{C}_4 \times 8 - {}^{11} \mathrm{C}_5 \times 2 \\\\ & = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} \times 8 - \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} \times 2 \\\\ & = (11 \times 10 \times 3) \times 8 - (11 \times 3 \times 2 \times 7) \times 2 \\\\ & = 330 \times 8 - 462 \times 2 \\\\ & = 2640 - 924 \\\\ & = 1716 \end{aligned}$

Now, let's examine how $1716$ can be expressed in the form of the options. The original solution cleverly factors the expression: $\begin{aligned} 1716 &= (11 \times 3 \times 4) \times (20 - 7) \quad \text{(This step seems to be a common factor extraction)} \end{aligned}$ Let's see if we can find a common factor. Coefficient of $x^{10} = {}^{11}C_4 \cdot 8 = \frac{11 \cdot 10 \cdot 9 \cdot 8}{4 \cdot 3 \cdot 2 \cdot 1} \cdot 8 = (11 \cdot 10 \cdot 3) \cdot 8 = 330 \cdot 8 = 2640$ Coefficient of $x^{7} = {}^{11}C_5 \cdot 2 = \frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot 2 = (11 \cdot 3 \cdot 2 \cdot 7) \cdot 2 = 462 \cdot 2 = 924$

We need to work with the exact terms before multiplying by 8 and 2, to see if there's a common factor related to ${}^{11}C_r$. Recall the identity: ${}^{n}C_r = \frac{n-r+1}{r} {}^{n}C_{r-1}$. So, ${}^{11}C_5 = \frac{11-5+1}{5} {}^{11}C_4 = \frac{7}{5} {}^{11}C_4$.

Let's substitute this: $\text{Difference} = {}^{11}C_4 \cdot 8 - \left(\frac{7}{5} {}^{11}C_4\right) \cdot 2$ $= {}^{11}C_4 \cdot 8 - \frac{14}{5} {}^{11}C_4$ $= {}^{11}C_4 \left(8 - \frac{14}{5}\right)$ $= {}^{11}C_4 \left(\frac{40 - 14}{5}\right)$ $= {}^{11}C_4 \left(\frac{26}{5}\right)$ $= 330 \times \frac{26}{5} = 66 \times 26 = 1716$ This confirms the value.

Now, let's look at the options again and the original solution's manipulation. The original solution made a mistake in its initial reduction steps to get the $12^3 - 12$ form.

Let's re-evaluate the target $11^3 - 11$. $11^3 - 11 = 1331 - 11 = 1320$. This is not 1716.

Let's re-examine the original solution's final algebraic manipulation. $11 \times 3 \times 4[20-7] = 11 \times 12 \times 13 = 1716$ This calculation is correct. The next step in the original solution is: $(12-1) \times 12 \times (12+1) = 12(12^2-1) = 12^3-12$ This transformation means $11 \times 12 \times 13$ is being rewritten in terms of 12. $11 \times 12 \times 13 = (12-1) \times 12 \times (12+1) = 12 (12^2 - 1^2) = 12(144-1) = 12 \times 143 = 1716$. So, $12^3 - 12 = 1728 - 12 = 1716$.

Therefore, the result $1716$ is indeed equal to $12^3 - 12$. The correct option is (C), not (A) as stated in the problem. There must be a typo in the provided "Correct Answer: A". I will proceed with the calculation leading to $12^3 - 12$.

Let's structure the algebraic simplification to match the solution path. We have: ${}^{11}C_4 \times 8 - {}^{11}C_5 \times 2$ $= \left(\frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}\right) \times 8 - \left(\frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1}\right) \times 2$ $= (11 \times 10 \times 3) \times 8 - (11 \times 3 \times 2 \times 7) \times 2$ $= 330 \times 8 - 462 \times 2$ $= 2640 - 924$ $= 1716$

Now, to express $1716$ in the form of options. The form $12^3 - 12$ is what we found. Let's see if the initial factoring in the original solution is valid: $11 \times 10 \times 9 \times 8 \times 8 / 24 = 11 \times 10 \times 3 \times 8 = 2640$ $11 \times 10 \times 9 \times 8 \times 7 \times 2 / 120 = 11 \times 3 \times 2 \times 7 \times 2 = 924$

The factoring part in the provided solution: $11 \times 3 \times 4[20-7]$ Let's try to derive this from $2640 - 924$: $2640 - 924 = 1716$. $11 \times 12 \times 13 = 1716$. $11 \times 12 \times 13 = 11 \times (3 \times 4) \times 13$. How did they get $20-7$? It seems they performed some creative factoring. Let's use the identity ${}^nC_r - {}^nC_{r-1} = {}^{n+1}C_r$. This isn't directly applicable here.

Let's follow the solution's factoring directly: ${}^{11}C_4 \times 8 - {}^{11}C_5 \times 2$ $= \frac{11 \times 10 \times 9 \times 8}{24} \times 8 - \frac{11 \times 10 \times 9 \times 8 \times 7}{120} \times 2$ $= 11 \times 10 \times 3 \times 8 - \frac{11 \times 10 \times 9 \times 8 \times 7 \times 2}{120}$ $= 2640 - 924$ The original solution then writes: $= 11 \times 3 \times 4[20-7]$ This is equivalent to $11 \times 12 \times 13 = 1716$. And then: $= (12-1) \times 12 \times (12+1) = 12(12^2-1) = 12^3-12$

So the result is $12^3 - 12$. The given correct answer (A) is $11^3 - 11$. This is a mismatch. I should clearly state the calculated answer and its equivalence to option (C).

Rewritten Solution continues:

$\text{Absolute Difference} = |({}^{11}C_4 \times 8) - ({}^{11}C_5 \times 2)|$ $= |(330 \times 8) - (462 \times 2)|$ $= |2640 - 924|$ $= |1716|$ $= 1716$

Now, let's verify which option matches 1716. (A) $11^3 - 11 = 1331 - 11 = 1320$ (B) $13^3 - 13 = 2197 - 13 = 2184$ (C) $12^3 - 12 = 1728 - 12 = 1716$ (D) $10^3 - 10 = 1000 - 10 = 990$

Our calculated value, 1716, matches option (C) $12^3 - 12$.

• Explanation: After calculating the numerical value of the difference, we compare it with the numerical values of the given options to find the correct match. This step ensures that the final answer is presented in the required format.

Tips and Common Mistakes to Avoid:

• Careful with Exponents: Pay close attention when simplifying terms like $(x^2)^{11-r}$ and $\left(\frac{1}{x}\right)^r$. A common error is misapplying exponent rules, leading to an incorrect power of $x$.
• Sign Errors: When dealing with terms like $\left(\frac{1}{2x}\right)^r$, remember it's $(2x)^{-r}$, so the exponent applies to both 2 and $x$.
• Binomial Coefficient Calculation: Ensure accurate calculation of ${}^nC_r$. Using the factorial definition and canceling terms before multiplying can save time and reduce errors.
• Algebraic Simplification: Be methodical when factoring out common terms or manipulating the final expression to match the options. Sometimes, the options are presented in an algebraically equivalent form rather than a direct numerical value.
• Absolute Difference: Remember to take the absolute value if the order of subtraction results in a negative number.

Summary and Key Takeaway

This problem is a classic application of the Binomial Theorem. The key steps involve:

1. Deriving and simplifying the general term ($T_{r+1}$) of the binomial expansion.
2. Equating the power of $x$ in the general term to the desired powers to find the corresponding values of $r$.
3. Substituting these $r$ values back into the coefficient part of the general term to find the required coefficients.
4. Finally, calculating the absolute difference and simplifying it to match one of the given options. Mastering the general term formula and careful algebraic manipulation are essential for solving such problems efficiently and accurately.