1. **State the problem:** We have a survey with three professions: Business, Government Jobs, and Farming. Given counts for each category and overlaps, we need to find the number of people with all three professions and the total number surveyed. 2. **Given data:** - Business total: $60$ - Government Jobs total: $45$ - Farming total: $125$ - Business only: $27$ - Government jobs only: $15$ - Business and government jobs only: $10$ - Government jobs and farming only: $5$ 3. **Define variables:** Let $x$ be the number of people with all three professions. 4. **Use the principle of inclusion-exclusion for three sets:** $$ |B| = \text{Business only} + \text{Business and Government only} + \text{Business and Farming only} + x $$ $$ |G| = \text{Government only} + \text{Business and Government only} + \text{Government and Farming only} + x $$ $$ |F| = \text{Farming only} + \text{Business and Farming only} + \text{Government and Farming only} + x $$ 5. **Calculate unknowns:** We know Business only ($27$), Government only ($15$), Business and Government only ($10$), Government and Farming only ($5$), but Business and Farming only is unknown. Let Business and Farming only be $y$. 6. **From Business total:** $$ 60 = 27 + 10 + y + x \implies 60 = 37 + y + x \implies y + x = 23 $$ 7. **From Government total:** $$ 45 = 15 + 10 + 5 + x \implies 45 = 30 + x \implies x = 15 $$ 8. **Substitute $x=15$ into $y + x = 23$:** $$ y + 15 = 23 \implies y = 8 $$ 9. **From Farming total:** $$ 125 = \text{Farming only} + y + 5 + x $$ We know $y=8$, $x=15$, so: $$ 125 = \text{Farming only} + 8 + 5 + 15 = \text{Farming only} + 28 \implies \text{Farming only} = 97 $$ 10. **Calculate total number of people:** Sum all mutually exclusive groups: $$ \text{Total} = \text{Business only} + \text{Government only} + \text{Farming only} + \text{Business and Government only} + \text{Business and Farming only} + \text{Government and Farming only} + \text{All three} $$ $$ = 27 + 15 + 97 + 10 + 8 + 5 + 15 = 177 $$ **Final answers:** - Number of people with all three professions: $15$ - Total number of people surveyed: $177$