There are $\left(n+1\right)$  white similar balls and $\left(n+1\right)$  black balls of different size. No. of ways the balls can be arranged in a row so that adjacent balls are of different colours is

Number of white similar balls $=\left(n+1\right)$ 

Number of black different size balls $=\left(n+1\right)$

Arrangement should be done either of these ways

$BWBWBW\cdots \left(or\right)WBWBWB\cdots$

They can be arranged in 2 ways

Black balls can be arranged in $\left(n+1\right)!$  ways

White balls can be arranged in $\left(n+1\right)!$  ways

$\therefore$  Required number of ways $=2\times {\left[\left(n+1\right)!\right]}^2$

$=2{\left[\left(n+1\right)!\right]}^2$  ways