For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of…
For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $P\left(2 n-1, n^2-4 n\right)$ from the line $x+y=8$ is:
$\sqrt{2}$
$2 \sqrt{2}$
$5 \sqrt{2}$
$3 \sqrt{2}$
Solution
No. of terms in $(x+y)^{(2 n-3)} \Rightarrow\left[\begin{array}{c}(2 n-3+1) \\ (2 n-2)\end{array}\right].$
$\therefore$ sum of all coefficients $=2^{2 n-3}$
(Put $\mathrm{x}=\mathrm{y}=1$ )
$\therefore$ Arithmetic mean of all coefficients
$\begin{aligned}
& =\left(\frac{2^{2 n-3}}{2 n-2}\right)=16 \\
& \Rightarrow 2^{2 n-3}=2^5(n-1) \Rightarrow n=5 \\
& \therefore P\left(2 n-1, n^2-4 n\right)=(9,5)
\end{aligned}$
$\begin{aligned} & x+y=8 \\ & \therefore P M=\left|\frac{9+5-8}{\sqrt{2}}\right|=\frac{6}{\sqrt{2}}=\frac{3 \times 2}{\sqrt{2}}=3 \sqrt{2}\end{aligned}$
