Question:
The term independent of $x$ in the expansion of $(x- \frac {3} {x^2})^{18}$
The term independent of $x$ in the expansion of $(x- \frac {3} {x^2})^{18}$
Updated On: Jul 7, 2022
- $^{18}C_6 $
- $^{18}C_6 \, 3^6 $
- $^{18}C_{12} \, {3}^{-6} $
- $3^6$
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The Correct Option is B
Solution and Explanation
$T_{r+1} = \,^{18}c_{r}x^{18-r} \left(-\frac{3}{x^{2}}\right)^{r}$
$= \, ^{18}c_{r}x^{18-r-2r}\left(-3\right)^{r}$
$\therefore 18 - 3r = 0$
$\Rightarrow r = 6$
$\therefore$ reqd. term $=\, ^{18}c_{6}\left(-3\right)^{6} = \, ^{18}c_{6}\left(3\right)^{6}$.
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.