$Given:\:\:A=\{1,2\}$

Binary operation $ \large *$ is a function from $\{(1,1),(1,2),(2,1),(2,2)\}\rightarrow \{1,2\}$

Also given that 1 is identity element and 2 is inverse of 2.

$\Rightarrow\:\large 1 *1=1,\:\:1\large *2=2\large *1=2\:\:and\:\:2\large *2=1$

This is the only possible mapping for the given conditions.

$\therefore \:$ only one binary operation is possible.