To solve this problem, we need to fill in the missing values in the table based on the given conditions.
Here are the given conditions:
1. The total number of campers is 32.
2. Out of these, 22 campers swim.
3. 20 campers play softball.
4. 5 campers neither swim nor play softball.
Using these values, let's break down the problem step-by-step:
### Step 1: Total number of campers who either swim or play softball or both
Given that there are 5 campers who do neither swim nor play softball, the remaining campers must be accounted for either swimming, playing softball, or both. Therefore:
[tex]\[ \text{Number of campers who either swim, play softball, or both} = 32 - 5 = 27 \][/tex]
### Step 2: Relation between swimming and playing softball
We need to fill in the two-way table such that it satisfies these values:
[tex]\[
\begin{array}{|l|c|c|c|}
\hline
& \text{Softball} & \begin{tabular}{c}
\text{No} \\
\text{Softball}
\end{tabular} & \text{Total} \\
\hline
\text{Swim} & a & b & 22 \\
\hline
\text{Not Swim} & c & 5 & d \\
\hline
\text{Total} & 20 & e & 32 \\
\hline
\end{array}
\][/tex]
### Step 3: Calculate [tex]\(e\)[/tex]
To find [tex]\(e\)[/tex], which is the number of campers who do not play softball:
[tex]\[ e = 32 - 20 = 12 \][/tex]
### Step 4: Fill in [tex]\(d\)[/tex]
To find [tex]\(d\)[/tex], which is the number of campers who do not swim:
[tex]\[ d = b + 5 = 10 + 5 = 15 \][/tex]
### Step 5: Fill in the rest of the table using the logic and given constraints
1. From the total columns for gymnastics
- [tex]\(a + c = 20\)[/tex]
- [tex]\(b + 5 = 12\)[/tex]. So, [tex]\( b = 12 - 5 = 7 \)[/tex].
2. From the rows for swimming
- [tex]\(a + b = 22\)[/tex]. We already determined [tex]\(b = 7\)[/tex], so:
[tex]\[ a = 22 - 7 = 15 \][/tex]
3. Finally calculate - [tex]\(a+c\)[/tex]
- Considering [tex]\(a + c = 20\)[/tex]
Button [tex]\(a\)[/tex], which we have found as 15:
[tex]\( c = 20 - 15 = 5 \)[/tex]
### Completing the Table
We have all the constraints fulfilled as:
[tex]\[
\begin{array}{|l|c|c|c|}
\hline
& \text{Softball} & \text{No Softball} & \text{Total} \\
\hline
\text{Swim} & 15 & 7 & 22 \\
\hline
\text{Not Swim} & 5 & 7 & 12 \\
\hline
\text{Total} & 20 & \ \ ( 32 - 20) \ = = 12 = & 32 \\
\hline
\end{array}
\][/tex]
### Answer Based on the above solution, the correct values are:
[tex]\[ a=15, b=7, c=5, d=10, e=12 \][/tex]
These fill the table while satisfying all the constraints.
So the correct option is:
[tex]\[ \boxed{a=15, b=7, c=5, d=10, e=12} \][/tex]
