Let's first convert all the mixed numbers into improper fractions and then perform the calculations step by step.
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Convert mixed numbers to improper fractions:
- \(18 \frac{1}{5} = \frac{18 \times 5 + 1}{5} = \frac{90 + 1}{5} = \frac{91}{5}\)
- \(1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5}\)
- \(2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}\)
- The whole number \(10\) can be expressed as \(\frac{10 \times 5}{5} = \frac{50}{5}\)
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Now substitute these back into the expression: \[ 18 \frac{1}{5} + 1 \frac{3}{5} - 2 \frac{1}{5} - 10 = \frac{91}{5} + \frac{8}{5} - \frac{11}{5} - \frac{50}{5} \]
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Combine these fractions: \[ \frac{91 + 8 - 11 - 50}{5} = \frac{91 + 8 - 11 - 50}{5} = \frac{91 + 8 = 99 - 11 = 88 - 50 = 38}{5} \]
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Now, we calculate: \[ \frac{38}{5} = 7 \frac{3}{5} \]
Now let's look at the options:
- \(17 \frac{2}{5}\)
- \( -10 \frac{1}{5} \)
- \(12 \frac{2}{5}\)
- \(7 \frac{3}{5}\)
The answer is \( \boxed{7 \frac{3}{5}} \).