There are 19 hockey players in a club. On a particular day, 14 were wearing the hockey shirts prescribed. None of them were without either hockey pants or shirts: Eleven were wearing the prescribed hockey pants. How many were in complete uniform?

Options

A. 6
B. 9
C. 7
D. 8

Explanation

To solve the problem step by step, we can use set theory to find out how many players were in complete uniform (i.e., wearing both the prescribed hockey shirts and pants).

### Step 1: Define the Sets
Let:
- Set A = Players wearing the prescribed hockey shirts
- Set B = Players wearing the prescribed hockey pants

From the problem, we know:
- Total players = 19
- Players wearing shirts (|A|) = 14
- Players wearing pants (|B|) = 11

### Step 2: Use the Formula for the Union of Two Sets
We need to find the number of players in both sets (|A ∩ B|), which represents the players wearing both shirts and pants. We can use the formula for the union of two sets:
\[ |A ∪ B| = |A| + |B| - |A ∩ B| \]
Where:
- |A ∪ B| is the total number of players wearing either shirts or pants.

### Step 3: Calculate |A ∪ B|
Since none of the players were without either hockey pants or shirts, the total number of players wearing either shirts or pants is equal to the total number of players:
\[ |A ∪ B| = 19 \]

### Step 4: Substitute the Known Values into the Formula
Now we can substitute the known values into the formula:
\[ 19 = 14 + 11 - |A ∩ B| \]

### Step 5: Solve for |A ∩ B|
Now, we can rearrange the equation to find |A ∩ B|:
\[ 19 = 25 - |A ∩ B| \]
\[ |A ∩ B| = 25 - 19 \]
\[ |A ∩ B| = 6 \]

### Conclusion
Therefore, the number of players who were in complete uniform (wearing both the prescribed hockey shirts and pants) is **6**.

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