### Instruction: Two sides of scalene $\bigtriangleup ABC$ measure $3$ centimeters and $5$ centimeters. How many different whole centimeter lengths are possible for the third side?### Response: To find the possible lengths for the third side of a scalene triangle, I need to recall the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.
This means that the third side of $\bigtriangleup ABC$ must be less than $3 + 5 = 8$ centimeters, but also more than $|3 - 5| = 2$ centimeters, since the difference of any two sides must be less than the third side.
So the possible values for the third side are $3, 4, 5, 6,$ and $7$ centimeters, since they are all whole numbers between $2$ and $8$.
However, I also need to remember that scalene means that no two sides are equal, so I have to eliminate $3$ and $5$ from the list, since they would make the triangle isosceles.
Therefore, the final answer is $3$, since there are only three different whole centimeter lengths that are possible for the third side of a scalene triangle with the given two sides.