Question

Assertion :The greatest coefficient in the expansion ${(1+7x)}^{10}$ is ${}^{10}{C}_5{7}^5$ Reason: The greatest coefficient in ${(1+x)}^{2n}$ is the middle term.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion,
• C. Assertion is true but Reason is false,
• D. Assertion is false but Reason is true.

Answer 1

The correct option is D Assertion is false but Reason is true.

For
$(1+7x{)}^{10}$, let $T_r$ denote the greatest term.
Therefore
$T_{r-1}<T_r>T_{r+1}$
Hence
${}^{10}{C}_{r-2}{7}^{r-2}<{}^{10}{C}_{r-1}{7}^{r-1}$
Hence
$\frac{10!}{(10-(r-2\left)\right)!(r-2)!}<\frac{10!}{(10-(r-1\left)\right)!.(r-1)!}.7$
$\frac{r-1}{10-r+2}<7$
$r-1<84-7r$
$8r<85$
$r<\frac{85}{8}$
$r<10.625$ ...(i)
$T_r>T_{r+1}$
${}^{10}{C}_{r-1}{7}^{r-1}>{}^{10}{C}_r{7}^r$
$\frac{10!}{(10-(r-1\left)\right)!(r-1)!}>\frac{10!}{(10-r)!.r!}7$
$\frac{r}{11-r}>7$
$r>77-7r$
$8r>77$
$r>\frac{77}{8}$
$r>9.625$
Hence
$9.625<r<10.625$
Hence $r=10$
The greatest term will be $T_{11}$
$={}^{10}{C}_{10}{7}^{10}$
$=7^{10}$