Correct Option (3)

The problem can be solved using principles of set theory. Let S represent the set of boys wearing hockey shirts and P represent the set of boys wearing hockey pants.

Total number of boys (N) = 19
Number of boys wearing shirts (|S|) = 11
Number of boys wearing pants (|P|) = 14

The statement "There are no boys without shirts and pants" implies that every boy is wearing at least a shirt or pants (or both). Therefore, the total number of boys is equal to the union of boys wearing shirts and boys wearing pants (|S ∪ P| = N).

The number of boys wearing full uniform corresponds to the intersection of boys wearing shirts and boys wearing pants (|S ∩ P|).

Using the Principle of Inclusion-Exclusion:

|S ∪ P| = |S| + |P| - |S ∩ P|

Substituting the given values:

19 = 11 + 14 - |S ∩ P|

19 = 25 - |S ∩ P|

|S ∩ P| = 25 - 19

|S ∩ P| = 6

Alternatively, consider the boys not wearing pants:

Number of boys not wearing pants = Total boys - Number of boys wearing pants = 19 - 14 = 5.

Since no boy is without shirts and pants, these 5 boys must be wearing only shirts.

Total number of boys wearing shirts = 11.
Number of boys wearing only shirts = 5.

Therefore, the number of boys wearing both shirts and pants (full uniform) is:

Number of boys in full uniform = (Total boys wearing shirts) - (Boys wearing only shirts)

Number of boys in full uniform = 11 - 5 = 6.

Incorrect Options:

Options 1 (3), 2 (5), and 4 (8) are incorrect because they do not satisfy the conditions derived from the problem statement. Any value other than 6 for the number of boys wearing full uniform would lead to a contradiction when applying the Principle of Inclusion-Exclusion or the alternative method. For example, if 5 boys were in full uniform, the total number of boys would be (11 - 5) + (14 - 5) + 5 = 6 (only shirts) + 9 (only pants) + 5 (both) = 20, which contradicts the given total of 19 boys.