Solution:

Provided

$n\text{ }\left( X\text{ }\cup \text{ }Y \right)\text{ }=\text{ }18$

$n\text{ }\left( X \right)\text{ }=\text{ }8$

$n\text{ }\left( Y \right)\text{ }=\text{ }15$

This can be written as

$n\text{ }\left( X\text{ }\cup \text{ }Y \right)\text{ }=\text{ }n\text{ }\left( X \right)\text{ }+\text{ }n\text{ }\left( Y \right)\text{ }\text{- }n\text{ }(X\cap Y)$

Now, substitute the values

$18\text{ }=\text{ }8\text{ }+\text{ }15\text{ }\text{- }n\text{ }(X\cap Y)$

Calculating further

$n\text{ }(X\cap Y)\text{ }=\text{ }23{-}18\text{ }=\text{ }5$

As a result, we get

$n\text{ }(X\cap Y)\text{ }=\text{ }5$