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:
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:
Show Hint
When dealing with expansions, simplify the general term first, then identify the required terms using the exponent relationships.
- \(10^3 - 10\)
- \(12^3 - 12\)
- \(11^3 - 11\)
- \(13^3 - 13\)
The Correct Option is B
Solution and Explanation
The general term in the expansion of \( \left( 2x^2 + \frac{1}{2x} \right)^{11} \) is:
\[ T_{r+1} = \binom{11}{r} (2x^2)^{11-r} \left( \frac{1}{2x} \right)^r. \]
Simplify:
\[ T_{r+1} = \binom{11}{r} 2^{11-r} x^{2(11-r)} \cdot \frac{1}{2^r x^r}. \]
\[ T_{r+1} = \binom{11}{r} \frac{2^{11-r}}{2^r} x^{22 - 3r}. \]
\[ T_{r+1} = \binom{11}{r} 2^{11-2r} x^{22 - 3r}. \]
For the coefficient of \( x^{10} \):
\[ 22 - 3r = 10 \implies r = 4. \]
The coefficient of \( x^{10} \) is:
\[ \binom{11}{4} 2^{11 - 2(4)} = \binom{11}{4} 2^{3}. \]
For the coefficient of \( x^7 \):
\[ 22 - 3r = 7 \implies r = 5. \]
The coefficient of \( x^7 \) is:
\[ \binom{11}{5} 2^{11 - 2(5)} = \binom{11}{5} 2^{1}. \]
Now calculate the absolute difference:
\[ \text{Difference} = \left| \binom{11}{4} 2^3 - \binom{11}{5} 2^1 \right|. \]
Expand the binomial coefficients:
\[ \binom{11}{4} = \frac{11 \cdot 10 \cdot 9 \cdot 8}{4 \cdot 3 \cdot 2 \cdot 1} = 330, \]
\[ \binom{11}{5} = \frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 462. \]
Substitute into the expression:
\[ \text{Difference} = \left| 330 \cdot 8 - 462 \cdot 2 \right|. \]
\[ \text{Difference} = \left| 2640 - 924 \right| = 1716. \]
Finally, express \(1716\) as \( 12^3 - 12 \):
\[ 12^3 = 1728, \quad 12^3 - 12 = 1728 - 12 = 1716. \]
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
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