Binomial Theorem
Grade 11
Question:
<p>If x<sup>m</sup> occurs in the expansion of (x + <span class="math-tex">\(\frac{1}{x^{2}}\)</span>)<sup>2n</sup>, then the coefficient of x<sup>m</sup> is</p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n) !}{(m) !(2 n-m) !}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n) ! 3 ! 3 !}{(2 n-m) !}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(3 n) !}{\left(\frac{2 n+m}{2}\right) !\left(\frac{4 n+m}{2}\right) !}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n) !}{\left(\frac{2 n-m}{3}\right) !\left(\frac{4 n+m}{3}\right) !}\)</span></p>
Step-by-Step Solution
Key Concept: The coefficient of $x^m$ is found by determining the general term $T_{r+1}$, equating the resulting power of $x$ to $m$, and solving for the specific index $r$.
<p>T<sub>r+1</sub> = <sup>2n</sup>C<sub>r</sub> x<sup>2n-r</sup> <span class="math-tex">$\left(\frac{1}{x^{2}}\right)^{r}$</span> = <sup>2n</sup>C<sub>r</sub> x<sup>2n-3r</sup>,<br />
This contains x<sup>m</sup>, if 2n - 3r = m i.e., if r = <span class="math-tex">$\frac{2 n-m}{3}$</span><br />
The coefficient of x<sup>m</sup> = <sup>2n</sup>C<sub>r</sub><br />
<span class="math-tex">$=\frac{(2 n) !}{(2 n-r) ! r !}=\frac{(2 n) !}{\left(2 n-\frac{2 n-m}{3}\right) !\left(\frac{2 n-m}{3}\right) !}$</span><br />
<span class="math-tex">$=\frac{(2 n) !}{\left(\frac{4 n+m}{3}\right) !\left(\frac{2 n-m}{3}\right) !}$</span></p>
Correct Answer: D