Binomial Theorem
Grade None
Question:
<p>The value of <span class="math-tex">\(-{ }^{15} \mathrm{C}_1+2 .{ }^{15} \mathrm{C}_2-3 .{ }^{15} \mathrm{C}_3+\ldots\)</span> <span class="math-tex">\( -15 .{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_1+{ }^{14} \mathrm{C}_3+{ }^{14} \mathrm{C}_5+\ldots +{ }^{14} \mathrm{C}_{11}\)</span> is:</p>
<p style="display:inline">2<sup>14</sup></p>
<p style="display:inline">2<sup>16</sup> – 1</p>
<p style="display:inline">2<sup>13</sup> – 14</p>
<p style="display:inline">2<sup>13</sup> – 13</p>
Step-by-Step Solution
<p><span class="math-tex">$\left(-{ }^{15} {C}_1+2 \cdot{ }^{15} {C}_2-3 \cdot{ }^{15} {C}_3+\ldots -15 \cdot{ }^{15} {C}_{15}\right)$</span> <span class="math-tex">$+\left({ }^{14} \mathrm{C}_1+{ }^{14} \mathrm{C}_3+\ldots .+{ }^{14} \mathrm{C}_{11}\right)$</span><br />
<span class="math-tex">$ =\sum\limits_{{r}=1}^{15}(-1)^{{r}} \cdot {r}^{15} {C}_{{r}}+\left({ }^{14} {C}_1+{ }^{14} {C}_3+\ldots+{ }^{14} {C}_{11}+{ }^{14} {C}_{13}\right)-{ }^{14} {C}_3 $</span><br />
<span class="math-tex">$ =\sum\limits_{{r}=1}^{15}(-1)^{{r}} 15 \cdot{ }^{14} {C}_{{r}-1}+2^{13}-14 $</span><br />
<span class="math-tex">$ =15\left(-{ }^{14} {C}_0+{ }^{14} {C}_1 \ldots \ldots . .-{ }^{14} {C}_{14}\right)+2^{13}-14 $</span><br />
<span class="math-tex">$ =2^{13}-14 $</span></p>
Correct Answer: C