A survey of 500 farmers showed that of the farmers, 124 grew only wheat, 110 grew only com, 86 grew only oats, 194 grew wheat, 59 grew wheat and com, 43 grew wheat and oats, and 179 grew com. Determine the number of farmers who a) grew at least one of the three, b) grew all three, c) did nof grow any of the three, and d) grew exactly fwo of the three. a) Or the farmers, \( \square \) grew at least one of the three crops. (Simpity your answer.) b) Or the farmers, \( \square \) grew all three crops. (Simplify your answer.) c) Or the larmers, \( \square \) did not grow any of the three crops. (Simplity your answer.) d) Of the farmers, \( \square \) grew exactiy two of the three cops. (Simplity your answer.)

To solve this problem, we will use the principle of inclusion-exclusion to find the number of farmers who grew at least one of the three crops, those who grew all three, those who did not grow any, and those who grew exactly two of the three crops. ### Known Conditions: - Total farmers surveyed: \( N = 500 \) - Farmers who grew only wheat: \( W = 124 \) - Farmers who grew only corn: \( C = 110 \) - Farmers who grew only oats: \( O = 86 \) - Farmers who grew wheat (including those who grew it with other crops): \( W_{total} = 194 \) - Farmers who grew both wheat and corn: \( W \cap C = 59 \) - Farmers who grew both wheat and oats: \( W \cap O = 43 \) - Farmers who grew corn (including those who grew it with other crops): \( C_{total} = 179 \) ### Step-by-Step Solution: 1. **Determine the number of farmers who grew all three crops**: Let \( x \) be the number of farmers who grew all three crops (wheat, corn, and oats). From the information given: - Farmers who grew wheat only: \( W = 124 \) - Farmers who grew corn only: \( C = 110 \) - Farmers who grew oats only: \( O = 86 \) - Farmers who grew wheat and corn but not oats: \( W \cap C - x = 59 - x \) - Farmers who grew wheat and oats but not corn: \( W \cap O - x = 43 - x \) - Farmers who grew corn and oats but not wheat: \( C \cap O - x \) (not directly given) We can express the total number of farmers who grew wheat as: \[ W + (W \cap C - x) + (W \cap O - x) + x = W_{total} \] Substituting the known values: \[ 124 + (59 - x) + (43 - x) + x = 194 \] Simplifying: \[ 124 + 59 + 43 - x = 194 \] \[ 226 - x = 194 \] \[ x = 32 \] 2. **Calculate the number of farmers who grew at least one of the three crops**: Using the inclusion-exclusion principle: \[ N_{at\ least\ one} = W + C + O + (W \cap C) + (W \cap O) + (C \cap O) - 2x \] We need to find \( C \cap O \). We can find it using the total number of corn farmers: \[ C_{total} = C + (W \cap C) + (C \cap O) - x \] Substituting the known values: \[ 179 = 110 + 59 + (C \cap O) - 32 \] \[ 179 = 137 + (C \cap O) \] \[ C \cap O = 42 \] Now we can calculate: \[ N_{at\ least\ one} = 124 + 110 + 86 + 59 + 43 + 42 - 2(32) \] \[ N_{at\ least\ one} = 124 + 110 + 86 + 59 + 43 + 42 - 64 \] \[ N_{at\ least\ one} = 400 \] 3. **Calculate the number of farmers who did not grow any of the three crops**: \[ N_{none} = N - N_{at\ least\ one} = 500 - 400 = 100 \] 4. **Calculate the number of farmers who grew exactly two of the three crops**: Farmers who grew exactly two crops can be calculated as: \[ N_{exactly\ two} = (W \cap C - x) + (W \cap O - x) + (C \cap O - x) \] Substituting the values: \[ N_{exactly\ two} = (59 - 32) + (43 - 32) + (42 - 32) \] \[ N_{exactly\ two} = 27 + 11 + 10 = 48 \] ### Final Answers: a) The number of farmers who grew at least one of the three crops: \( \boxed{400} \) b) The number of farmers who grew all three crops: \( \boxed{32} \) c) The number of farmers who did not grow any of the three crops: \( \boxed{100} \) d) The number of farmers who grew exactly two of the three crops: \( \boxed{48} \)