For an integer n ≥ 2 , if the arithmetic mean of all coefficients...

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}$$<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/zQOPegNaGroihIXObDobjfcSVyZC8NsiEeh7ci6YInA.original.fullsize.png"/><br/>$$\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}$$