Binomial Theorem
Grade 11
Question:
<p>The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 - x )<sup>100</sup>, is equal to</p>
<p style="display:inline"><sup>99</sup>C<sub>49</sub></p>
<p style="display:inline">-<sup>99</sup>C<sub>49</sub></p>
<p style="display:inline">-<sup>101</sup>C<sub>50</sub></p>
<p style="display:inline">-<sup>101</sup>C<sub>51</sub></p>
Step-by-Step Solution
Key Concept: Utilize the symmetry property of binomial coefficients $inom{n}{r} = inom{n}{n-r}$ and the fact that the alternating sum of all coefficients in the expansion of $(1-x)^n$ is zero.
<p>By Binomial expression<br />
(1 - x)<sup>100</sup> = C<sub>0</sub> - C<sub>1</sub>x + C<sub>2</sub>x<sup>2</sup> - ... C<sub>99</sub>x<sup>99</sup> + C<sub>100</sub>x<sup>100</sup><br />
<span class="math-tex">$\Rightarrow$</span> C<sub>0</sub> - C<sub>1</sub> + C<sub>2</sub> - C<sub>3</sub> + ... -C<sub>99</sub> + C<sub>100</sub> = 0<br />
2(C<sub>0</sub> - C<sub>1</sub> + C<sub>2</sub> + ... -C<sub>9</sub> + C<sub>50 </sub>= 0<br />
C<sub>0</sub> - C<sub>1</sub> + C<sub>2</sub> + ...C<sub>99</sub> = <span class="math-tex">$-\frac{1}{2}$</span>C<sub>50</sub><br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$-\frac{1}{2} \frac{100 !}{50 ! 50 !}$</span> = <span class="math-tex">$-\frac{1}{2} \times \frac{100 ! \times 99 !}{50 ! 50 !}$</span> = -<sup>99</sup>C<sub>49</sub></p>
Correct Answer: B