A survey of 500 farmers showed that of the​ farmers, 122 grew only​ wheat, 106 grew only​ corn, 91 grew only​ oats, 197 grew​ wheat, 60 grew wheat and​ corn, 54 grew wheat and​ oats, and 174 grew corn. Determine the number of farmers who ​a) grew at least one of the​ three, ​b) grew all​ three, ​c) did not grow any of the​ three, and ​d) grew exactly two of the three.


We are given a survey of 500 farmers growing wheat (W), corn (C), and oats (O). From the information provided:

- Farmers who grew only wheat: 122
- Farmers who grew only corn: 106
- Farmers who grew only oats: 91
- Farmers who grew wheat: 197
- Farmers who grew corn: 174
- Farmers who grew wheat and corn: 60
- Farmers who grew wheat and oats: 54

Let’s determine each part of the question step by step.

### Step 1: Defining the sets
- Let \( W \) represent the set of farmers growing wheat.
- Let \( C \) represent the set of farmers growing corn.
- Let \( O \) represent the set of farmers growing oats.

We are given some of these directly, while others will require calculations.

### Part (a) Farmers who grew at least one of the three crops

To find the number of farmers who grew at least one of the crops, we can use the principle of inclusion-exclusion:

\[
|W \cup C \cup O| = |W| + |C| + |O| - |W \cap C| - |W \cap O| - |C \cap O| + |W \cap C \cap O|
\]

We already know:
- \( |W| = 197 \)
- \( |C| = 174 \)
- \( |O| \) is the total number of oat growers. Since 91 farmers grow only oats and 54 grow wheat and oats, the total number of oat growers is \( 91 + 54 = 145 \).
- \( |W \cap C| = 60 \)
- \( |W \cap O| = 54 \)

We don’t know \( |C \cap O| \) and \( |W \cap C \cap O| \) yet, so we will need to find them in later steps.

### Part (b) Farmers who grew all three crops

To determine the number of farmers who grew all three crops (wheat, corn, and oats), we can use the fact that:

\[
|W \cup C| = \text{Number of farmers who grew both wheat and corn} - \text{those who grew all three}
\]

We will return to this after calculating missing intersections.

### Part (c) Farmers who did not grow any of the three crops

To find the number of farmers who did not grow any of the crops, we subtract the number of farmers who grew at least one crop from the total number of farmers:

\[
\text{Farmers who did not grow any} = 500 - |W \cup C \cup O|
\]

### Part (d) Farmers who grew exactly two of the three crops

The number of farmers who grew exactly two crops is determined by subtracting those who grew all three from the number who grew each combination of two crops.

This includes:
- \( |W \cap C| \) minus those who grew all three
- \( |W \cap O| \) minus those who grew all three
- \( |C \cap O| \), which we will calculate

We will solve this step-by-step after addressing unknown values in Part (b).

Let me continue and provide the exact solution for each part using Python calculations.The solution shows that the total number of farmers who grew at least one of the three crops depends on the number of farmers who grew both corn and oats. Specifically, the formula indicates that:

\[
|W \cup C \cup O| = \text{{corn and oats}} + 98
\]

Since the total number of farmers is 500, we can use this relation to find the number of farmers who grew all three crops.

Next, I will determine the value of \( |W \cap C \cap O| \) and \( |C \cap O| \) to finalize the solution for each part.It seems there was a calculation error leading to the result \( |W \cup C \cup O| = 500 \), which is incorrect based on our problem. Let me correct this and solve it accurately.

I'll rework the calculation.It appears my previous correction led to a redundant equation. I will now recalculate using a simplified approach by solving the key variables directly to avoid further issues. Let's start by clearly isolating the intersections and apply them to the inclusion-exclusion formula properly for accurate results.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can walk you through how to approach this manually, step by step. Would you like that?

A survey of 500 farmers showed that of the farmers, 122 grew only wheat, 106 grew only corn, 91 grew only oats, 197 grew wheat, 60 grew wheat and corn, 54 grew wheat and oats, and 174 grew corn. Determine the number of farmers who a) grew at least one of the three, b) grew all three, c) did not grow any of the three, and d) grew exactly two of the three.

Solve a set theory problem based on a survey of 500 farmers growing wheat, corn, and oats. Learn how to use the Inclusion-Exclusion Principle to find the number of farmers growing at least one crop, all three, or none of the crops. Suitable for Grades 10-12.

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