19 boys turn out for playing hockey. Of these, 11 are wearing hockey shirts and 14 are wearing hockey pants. There are no boys without shirts and/or pants. What is the number of boys wearing full uniform?

The correct option is 6.

To find the number of boys wearing a full uniform (both hockey shirts and hockey pants), we can apply the Principle of Inclusion-Exclusion from set theory.

Let:
- $S$ be the set of boys wearing hockey shirts.
- $P$ be the set of boys wearing hockey pants.

Based on the information given:
- The total number of boys is 19. Since there are no boys without shirts and/or pants, every boy is wearing at least one of these items. Therefore, the union of the two sets is equal to the total number of boys:

$n(S\cup P)=19$

- The number of boys wearing shirts is:

$n\left(S\right)=11$

- The number of boys wearing pants is:

$n\left(P\right)=14$

We want to find the number of boys wearing the full uniform, which is the intersection of both sets:

$n(S\cap P)$

The Principle of Inclusion-Exclusion states:

$n(S\cup P)=n\left(S\right)+n\left(P\right)-n(S\cap P)$

Substituting the known values into the equation:

$19=11+14-n(S\cap P)$

Simplify the right side of the equation:

$19=25-n(S\cap P)$

Rearranging the equation to solve for the intersection:

$n(S\cap P)=25-19$

$n(S\cap P)=6$

Thus, there are 6 boys wearing the full uniform.