Binomial Theorem
Grade None
Question:
<p>The greatest coefficient in the expansion of (1 + x)<sup>2n+2</sup> is</p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n+2) !}{n !(n+1) !}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n) !}{n !(n+1) !}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n+2) !}{\{(n+1) !\}^{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{(2 n) !}{(n !)^{2}}\)</span></p>
Step-by-Step Solution
Key Concept: For a binomial expansion (1+x)^N where N is an even integer, the greatest coefficient is the coefficient of the middle term, calculated as nCr where r = N/2.
<p>Greatest coefficient of (1 + x)<sup>2n+2</sup><br />
= <sup>(2n+2)</sup>C<sub>n+1</sub> = <span class="math-tex">$\frac{(2 n+2) !}{\{(n+1) !\}^{2}}$</span></p>
Correct Answer: C