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counting and probability intermediate

Problem

Joy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
(A)
(B)
(C)
(D)
Solution
The triangle inequality generalizes to all polygons, so and yields . Now, we know that there are numbers between and exclusive, but we must subtract to account for the 2 lengths already used that are between those numbers, which gives
Final answer
B