$(4$ points$)$ A highly composite number $($also known as an anti$-$prime$)$ is a positive integer with more factors than any smaller positive integer has. For example, $12$ is highly composite because it has six factors $(1,2,3,4,6,12),$ which surpasses the number of factors of each integer from $1$ to $11,$ inclusive.

Prove that there are exactly three highly composite numbers between $2$ and $11,$ inclusive.