How many non-[congruent](https://artofproblemsolving.com/wiki/index.php/Congruent_(geometry)) [triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) with [perimeter](https://artofproblemsolving.com/wiki/index.php/Perimeter) $7$ have [integer](https://artofproblemsolving.com/wiki/index.php/Integer) side lengths? 
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$
By the [triangle inequality](https://artofproblemsolving.com/wiki/index.php/Triangle_inequality), no side may have a length greater than the semiperimeter, which is $\frac{1}{2}\cdot7=3.5$. 
Since all sides must be integers, the largest possible length of a side is $3$. Therefore, all such triangles must have all sides of length $1$, $2$, or $3$. Since $2+2+2=6<7$, at least one side must have a length of $3$. Thus, the remaining two sides have a combined length of $7-3=4$. So, the remaining sides must be either $3$ and $1$ or $2$ and $2$. Therefore, the number of triangles is $\boxed{\mathrm{(B)}\ 2}$.