To determine how many different triangles Joanna can make using the four straws of lengths 3 inches, 7 inches, and 10 inches, we first need to make use of the triangle inequality theorem, which states that for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Given the lengths of the straws: 3 inches, 7 inches, and 10 inches, we'll analyze if any combination of them can form a triangle.
Joanna needs to choose 3 straws from the available 4 straws. The combinations of 3 straws she can choose from 4 can be calculated as:
\[ \text{Combinations} = \binom{4}{3} = 4 \]
The combinations of 3 straws are:
- 3, 7, 10
- 3, 7, (another straw, if any)
- 3, 10, (another straw, if any)
- 7, 10, (another straw, if any)
Since there are only three actual lengths (3, 7, and 10), we only need to consider the combination of 3, 7, and 10.
Let’s check if the lengths 3 inches, 7 inches, and 10 inches satisfy the triangle inequality:
- Check \(3 + 7 > 10\): \[ 10 > 10 \quad \text{(not satisfied)} \]
Since the first condition fails, we can conclude that the lengths 3, 7, and 10 cannot form a triangle because they do not satisfy the triangle inequality.
Since there are no other combinations (as we can’t combine with a non-existent fourth straw), we conclude that:
Joanna cannot make any triangles with the given lengths.
Thus, the total number of different triangles Joanna can make is 0.
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