How many differently shaped triangles exist in which no two sides are of thesame length, each side is of integral unit length and the perimeter of the triangleis less than 14 units

Sure, let's solve this step by step.

Step 1: Understand the problem We are looking for triangles with no two sides of the same length, each side is of integral unit length, and the perimeter of the triangle is less than 14 units.

Step 2: Apply the triangle inequality theorem The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Step 3: Find the possible combinations Since the perimeter is less than 14 units, the maximum length for a side of the triangle is 13 units. However, because no two sides can be the same length, the maximum length for a side is actually less than 13 units.

Let's start with the smallest possible triangle, which would have sides of 1, 2, and 3 units (since no two sides can be the same length).

From there, we can incrementally increase the lengths of the sides, making sure to follow the triangle inequality theorem and the condition that the perimeter is less than 14 units.

For example, the next triangle could have sides of 1, 2, and 4 units. Then 1, 3, and 4 units. Then 1, 3, and 5 units, and so on.

Step 4: Count the combinations Continue this process until you have found all possible combinations.

Remember, the order of the sides does not matter (i.e., a triangle with sides of 1, 2, and 3 units is the same as a triangle with sides of 3, 2, and 1 units), so don't count the same triangle more than once.

This process will give you the total number of differently shaped triangles that meet the given conditions.