Binomial Theorem
Grade 11
Question:
<p>Let S be the sum of the digits of the coefficient of x<sup>6</sup> in the expansion of (1 + 2x - 3x<sup>2</sup>)<sup>4</sup>. Then which of the following statements is not correct?</p>
<p style="display:inline">S is divisible by 9</p>
<p style="display:inline">G. C. D. of Sand 6 is 6</p>
<p style="display:inline">S is the square of an integer</p>
<p style="display:inline">S is divisible by 3</p>
Step-by-Step Solution
Key Concept: The coefficient of a term in a multinomial expansion is found by identifying all sets of non-negative integers $(p, q, r)$ that satisfy $p+q+r=n$ and yield the target exponent in the general term formula.
<p>The general term of the expansion is<br />
<span class="math-tex">$\frac{4 !}{p ! q ! r !}$</span> <span class="math-tex">$\cdot$</span> 1<sup>p</sup> (2x)<sup>q</sup> (-3x<sup>2</sup>)<sup>r</sup>, where p + q + r = 4<br />
<span class="math-tex">$=\frac{4 !}{p ! q ! r !}$</span> (2)<sup>q</sup> (-3)<sup>r</sup> x<sup>q+2r</sup><br />
We require p, q, r such that<br />
p + q + r = 4 and q + 2r = 6<br />
<span class="math-tex">$\Rightarrow$</span> p = 1, q = 0 and r = 3<br />
or p = 0, q = 2 and r = 2<br />
<span class="math-tex">$\Rightarrow$</span> The coefficient of x<sup>6</sup><br />
<span class="math-tex">$=\frac{4 !}{1 ! 0 ! 3 !}$</span> 2<sup>0</sup> (-3)<sup>3</sup> + <span class="math-tex">$\frac{4 !}{0 ! 2 ! 2 !}$</span> (2)<sup>2</sup> (-3)<sup>2</sup><br />
= -108 + 216 = 108<br />
<span class="math-tex">$\Rightarrow$</span> S = 9<br />
S is the perfect square of an integer (true)<br />
S is divisible by 3 and 9 (true)<br />
S and 6 have G. C. D. = 3</p>
Correct Answer: B