2003 AMC 12A Problem 7

Let the sides be $a\le b\le c$ with $a+b+c=7.$ The triangle inequality requires $c\lt a+b,$ so $c\lt3.5,$ forcing $c=3.$ Then $a+b=4$ with $a\le b\le3,$ giving the triangles $1\text{-}3\text{-}3$ and $2\text{-}2\text{-}3.$ Thus, the correct answer is B.