Binomial Theorem
Grade None
Question:
<p>If the coefficient of 4<sup>th</sup> term in the expansion of (a + b)<sup>n</sup> is 56, then n is</p>
<p style="display:inline">10</p>
<p style="display:inline">8</p>
<p style="display:inline">12</p>
<p style="display:inline">6</p>
Step-by-Step Solution
Key Concept: The coefficient of the k-th term in the binomial expansion of $(a+b)^n$ is determined by the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$, where $r = k-1$.
<p>T<sub>4</sub> = T<sub>3+1</sub> = <sup>n</sup>C<sub>3</sub> a<sup>n-3</sup> b<sup>3</sup><br />
According to the given condition,<br />
<sup>n</sup>C<sub>3</sub> = 56<br />
<span class="math-tex">\(\Rightarrow \frac{n !}{3 !(n-3) !}\)</span> = 56<br />
<span class="math-tex">\(\Rightarrow\)</span> n(n - 1) (n - 2) = 56 <span class="math-tex">\(\times\)</span> 6<br />
<span class="math-tex">\(\Rightarrow\)</span> n(n - 1) (n - 2) = 8 <span class="math-tex">\(\times\)</span> 7 <span class="math-tex">\(\times\)</span> 6<br />
<span class="math-tex">\(\Rightarrow\)</span> n = 8</p>
Correct Answer: B