### Instruction: A triangle can be formed having side lengths $4,$ $5,$ and $8.$ It is impossible, however, to construct a triangle with side lengths $4,$ $5,$ and $10.$ Using the side lengths $2,$ $3,$ $5,$ $7,$ and $11,$ how many different triangles with exactly two equal sides can be formed?### Response: This problem is about isosceles triangles, which have two equal sides and one unequal side.
To form an isosceles triangle, I need to choose two equal sides and one unequal side from the given lengths.
I can use the triangle inequality to check if a given combination of sides can form a triangle.
The triangle inequality says that the sum of any two sides of a triangle must be greater than the third side.
For example, if I choose $2,$ $2,$ and $3$ as the sides, I can form a triangle because $2+2 > 3,$ $2+3 > 2,$ and $3+2 > 2.$
But if I choose $2,$ $2,$ and $5,$ I cannot form a triangle because $2+2 < 5.$
I can list all the possible combinations of two equal sides and one unequal side, and then eliminate those that violate the triangle inequality.
The possible combinations are: 
$2, 2, 3$
$2, 2, 5$
$2, 2, 7$
$2, 2, 11$
$3, 3, 2$
$3, 3, 5$
$3, 3, 7$
$3, 3, 11$
$5, 5, 2$
$5, 5, 3$
$5, 5, 7$
$5, 5, 11$
$7, 7, 2$
$7, 7, 3$
$7, 7, 5$
$7, 7, 11$
$11, 11, 2$
$11, 11, 3$
$11, 11, 5$
$11, 11, 7$
The combinations that violate the triangle inequality are: 
$2, 2, 5$
$2, 2, 7$
$2, 2, 11$
$3, 3, 11$
$5, 5, 2$
$5, 5, 3$
$7, 7, 2$
$7, 7, 3$
$11, 11, 2$
$11, 11, 3$
$11, 11, 5$
$11, 11, 7$
So, I can eliminate these 12 combinations from the list.
That leaves me with 8 valid combinations that can form isosceles triangles.
Therefore, the answer is 8.