Question

The interval in which $x$ must lie so that the numerically greatest term in the expansion of ${(1-x)}^{21}$ has the greatest coefficient is $(\frac{a}{b},\frac{6}{5})$
Find the value of $a+b$

• A. 8
• B. 9
• C. 10
• D. 11

Answer 1

The correct option is D 11

If $n$ is odd, then numerically greatest coefficient in the expansion of
${(1-x)}^n$ is $\frac{{}^n{C}_{n-1}}{2}$ or $\frac{{}^n{C}_{n+1}}{2}$
Therefore in ${(1-x)}^{21},$ the numerically greatest coefficient is ${}^{21}{C}_{10}$ or ${}^{21}{C}_{11}.$

So, the numerically greatest term
${=}^{21}{C}_{11}{x}^{11}$ or ${}^{21}{C}_{10}{x}^{10}$ and $\left|{}^{21}{C}_{10}{x}^{10}\right|>\left|{}^{21}{C}_9.x^9\right|$
$\frac{21!}{10!11!}>\frac{21!}{9!10!}x$ and $\frac{21!}{11!10!}x>\frac{21!}{9!12!}(\because x>0)$
$\Rightarrow x<\frac{6}{5}$ and $x>\frac{5}{6}\Rightarrow x\in (\frac{5}{6},\frac{6}{5})$