Binomial Theorem
Grade 11
Question:
<p>The sum of the coefficients in the expansion of (x + y)<sup>n</sup> is 4096. The greatest coefficient in the expansion is</p>
<p style="display:inline">724</p>
<p style="display:inline">1024</p>
<p style="display:inline">824</p>
<p style="display:inline">924</p>
Step-by-Step Solution
Key Concept: The sum of binomial coefficients in the expansion of $(x+y)^n$ is $2^n$, and for an even $n$, the greatest binomial coefficient is the central term ${}^nC_{n/2}$.
<p>Sum of coefficients of (x +y)<sup>n</sup> = 2<sup>n</sup><br />
<span class="math-tex">\(\Rightarrow\)</span> 2<sup>n</sup> = 4096 = 2<sup>12</sup><br />
<span class="math-tex">\(\therefore\)</span> n = 12<br />
Since n is even, hence greatest coefficient<br />
= <span class="math-tex">\({ }^{n} \mathrm{C}_{\frac{n}{2}}={ }^{12} \mathrm{C}_{6}=\frac{12.11 .10 .9 .8 .7}{1.2 .3 .4 .5 .6}\)</span> = 924</p>
Correct Answer: D