Part III.

Check whether the set $S=R-\{-1\}$ is a group under the binary operation $*$ defined as

$a*b=a+b+ab$ for any $a,$binS.

Express the connection between the order of a group and the quantity of binary operations that

can be defined on that set. Apply to answer the question "What is the total number of binary

operations that can be defined on a set containing $3$ elements?"