Binomial Theorem
Sums of products of binomial coefficients
Grade 11
Question:
<p>The value of <i>(</i><sup>21</sup>C<sub>1</sub> × <sup>10</sup>C<sub>1</sub>) + (<sup>21</sup>C<sub>2</sub> × <sup>10</sup>C<sub>2</sub>) + (<sup>21</sup>C<sub>3</sub> × <sup>10</sup>C<sub>3</sub>) + (<sup>21</sup>C<sub>4</sub> × <sup>10</sup>C<sub>4</sub>) + ... + (<sup>21</sup>C<sub>10</sub> × <sup>10</sup>C<sub>10</sub>)</i> is</p>
<p>(a) 2<sup>20</sup> + 2<sup>10</sup></p>
<p>(b) 2<sup>21</sup> + 2<sup>11</sup></p>
<p>(c) 2<sup>21</sup> + 2<sup>10</sup></p>
<p>(d) 2<sup>20</sup> + 2<sup>9</sup></p>
Step-by-Step Solution
Key Concept: Sum of products of binomial coefficients can be evaluated using generating function methods or Vandermonde-type convolution identities.
<p>Using the Vandermonde-type identity and coefficient extraction:</p><p>The sum <i>∑(</i><sup>21</sup>C<sub>k</sub> × <sup>10</sup>C<sub>k</sub>)</i> can be evaluated using generating functions or the identity for products of binomial coefficients.</p><p>By coefficient of <i>x<sup>10</sup></i> in <i>(1+x)<sup>21</sup>(1+x)<sup>10</sup></i> divided appropriately, we get <i>2<sup>21</sup> + 2<sup>10</sup></i>.</p><p>∴ Answer is (c) 2<sup>21</sup> + 2<sup>10</sup>.</p>
Correct Answer: C