Binomial Theorem
Grade 11
Question:
<p>The coefficient of x<sup>4</sup> in the expansion of (1 + x + x<sup>2 </sup>+ x<sup>3</sup>)<sup>11</sup> is</p>
<p style="display:inline">330</p>
<p style="display:inline">605</p>
<p style="display:inline">660</p>
<p style="display:inline">990</p>
Step-by-Step Solution
Key Concept: Factorize the multinomial into a product of binomials to simplify the problem into finding all combinations of terms from each expansion whose exponents sum to the target power.
<p>(1 + x + x<sup>2</sup> + x<sup>3</sup>)<sup>11</sup> = (1 + x)<sup>11</sup> (1 + x<sup>2</sup>)<sup>11</sup><br />
x<sup>4</sup> appears as product of (x<sup>2</sup>, x<sup>2</sup>) or (x, x, x<sup>2</sup>) or (x, x, x, x)<br />
<span class="math-tex">$\Rightarrow$</span> The coefficient of x<sup>4</sup><br />
= <sup>11</sup>C<sub>2</sub> + <sup>11</sup>C<sub>2</sub> <sup>11</sup>C<sub>1</sub> + <sup>11</sup>C<sub>4</sub><br />
= 55 + 11(55) + 330 = 990</p>
Correct Answer: D