Binomial Theorem
Grade 11
Question:
<p>C<sub>0</sub> - C<sub>1</sub> + C<sub>2</sub> - C<sub>3</sub> + ... + (-1)<sup>n</sup> C<sub>n</sub> is equal to</p>
<p style="display:inline">0</p>
<p style="display:inline">2<sup>n</sup></p>
<p style="display:inline">2<sup>n-1</sup></p>
<p style="display:inline">2<sup>n</sup> - 1</p>
Step-by-Step Solution
Key Concept: The alternating sum of binomial coefficients is evaluated by substituting x = -1 into the binomial expansion of (1 + x)^n, which always results in zero.
<p>We know that,<br />
(1 + x)<sup>n</sup> = <sup>n</sup>C<sub>0</sub> + <sup>n</sup>C<sub>1</sub>x + <sup>n</sup>C<sub>2</sub>x<sup>2</sup> + ... + <sup>n</sup>C<sub>n</sub>x<sup>n</sup><br />
Substituting x = -1, we get<br />
(1 - 1)<sup>n</sup> = <sup>n</sup>C<sub>0</sub> - <sup>n</sup>C<sub>1</sub> + <sup>n</sup>C<sub>2</sub> - ... (-1)<sup>n</sup> <sup>n</sup>C<sub>n</sub><br />
<span class="math-tex">$\therefore$</span> C<sub>0</sub> - C<sub>1</sub> + C<sub>2</sub> - C<sub>3 </sub>+ ... (-1)<sup>n</sup> C<sub>n</sub> = 0</p>
Correct Answer: A