Binomial Theorem
Grade 11
Question:
<p>If A and B are the coefficients of x<sup>n</sup> in the expansions of (1 + x)<sup>2n</sup> and (1 + x)<sup>2n-1</sup> respectively, then</p>
<p style="display:inline">A <span class="math-tex">\(\ne\)</span> B</p>
<p style="display:inline">A = 2B</p>
<p style="display:inline">2A = B</p>
<p style="display:inline">A = B</p>
Step-by-Step Solution
Key Concept: Identify the coefficients of x^n as binomial combinations and simplify their ratio using the factorial formula.
<p><span class="math-tex">$\frac{\text { Coefficient of } x^{n} \text { in expansion of }(1+x)^{2 n}}{\text { Coefficient of } x^{n} \text { in expansion of }(1+x)^{2 n-1}}$</span><br />
<span class="math-tex">$=\frac{{ }^{2 n} \mathrm{C}_{n}}{{ }^{(2 n-1)} \mathrm{C}_{n}}=\frac{(2 n) !}{n ! n !} \times \frac{(n-1) ! n !}{(2 n-1) !}=\frac{2 n}{n}$</span> = 2<br />
<span class="math-tex">$\Leftrightarrow \frac{A}{B}=\frac{2}{1} \Leftrightarrow$</span> A = 2B</p>
Correct Answer: B