1. The problem is to evaluate the expression $$2\frac{1}{3} + 4\frac{3}{1} - 3\frac{1}{1} \times 5\frac{2}{1} \div 7\frac{4}{1} + 6\frac{2}{1}$$ where mixed numbers are given. 2. Convert all mixed numbers to improper fractions: $$2\frac{1}{3} = \frac{7}{3}, \quad 4\frac{3}{1} = \frac{7}{1}, \quad 3\frac{1}{1} = \frac{4}{1}, \quad 5\frac{2}{1} = \frac{7}{1}, \quad 7\frac{4}{1} = \frac{11}{1}, \quad 6\frac{2}{1} = \frac{8}{1}$$ 3. Substitute these into the expression: $$\frac{7}{3} + \frac{7}{1} - \frac{4}{1} \times \frac{7}{1} \div \frac{11}{1} + \frac{8}{1}$$ 4. Perform multiplication and division from left to right: $$\frac{4}{1} \times \frac{7}{1} = \frac{28}{1}$$ $$\frac{28}{1} \div \frac{11}{1} = \frac{28}{11}$$ 5. Now the expression is: $$\frac{7}{3} + \frac{7}{1} - \frac{28}{11} + \frac{8}{1}$$ 6. Find a common denominator for all fractions, which is 33: $$\frac{7}{3} = \frac{77}{33}, \quad \frac{7}{1} = \frac{231}{33}, \quad \frac{28}{11} = \frac{84}{33}, \quad \frac{8}{1} = \frac{264}{33}$$ 7. Substitute and combine: $$\frac{77}{33} + \frac{231}{33} - \frac{84}{33} + \frac{264}{33} = \frac{77 + 231 - 84 + 264}{33} = \frac{488}{33}$$ 8. Simplify if possible. Since 488 and 33 share no common factors other than 1, the fraction is in simplest form. 9. Convert to mixed number: $$488 \div 33 = 14 \text{ remainder } 26$$ So, $$\frac{488}{33} = 14\frac{26}{33}$$ Final answer: $$14\frac{26}{33}$$