To determine how many different triangles Joanna can make with combinations of three straws, we need to apply the triangle inequality theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following must hold true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
This means that the sum of the lengths of any two sides must be greater than the length of the third side.
Although you did not provide specific lengths of the straws, let's consider a hypothetical situation where Joanna has straws of different lengths represented as a set of integers. For the sake of explanation, let's assume the straw lengths are as follows:
- Lengths: \( 1, 2, 3, 4, 5, 6 \)
Now, we will analyze how Joanna can form triangles using combinations of these lengths.
- Create all possible combinations of three straws: \( {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, ... \).
- For each combination, verify the triangle inequality.
Let's evaluate combinations:
- \( {1, 2, 3} \):
- \( 1 + 2 = 3\) (not valid, equal)
- \( {1, 2, 4} \):
- \( 1 + 2 < 4\) (not valid)
- \( {1, 2, 5} \):
- \( 1 + 2 < 5\) (not valid)
- \( {1, 2, 6} \):
- \( 1 + 2 < 6\) (not valid)
- \( {1, 3, 4} \):
- \( 1 + 3 = 4\) (not valid)
- \( {1, 3, 5} \):
- \( 1 + 3 < 5\) (not valid)
- \( {1, 3, 6} \):
- \( 1 + 3 < 6\) (not valid)
- Continue this process for other combinations...
Finally, after checking all combinations, the valid triangles that can be formed would be those that satisfy the triangle inequality.
Assuming Joanna uses the 5 straw lengths \( 2, 3, 4, 5, 6 \) instead:
- The triangular combinations would likely be more valid. For instance:
- \( {2, 3, 4}\): \( 2 + 3 > 4 \) (valid)
- \( {2, 4, 5}\): \( 2 + 4 > 5 \) (valid)
- \( {3, 4, 5}\): \( 3 + 4 > 5 \) (valid)
- \( {3, 5, 6}\): \( 3 + 5 > 6 \) (valid)
Counting all valid combinations would yield (let’s say 5 valid triangles).
To summarize, without the specific lengths of the straws Joanna is using, I can't give an exact number. However, once the lengths are known, follow the above steps to validate each combination systematically and determine the count of valid triangles.