NO VENN DIAGRAMS!

We decompose set X as follows:

$X$

$=X \cap U$

$= (X \cap U) \cap U$

$= [X \cap (Y \cup Y')] \cap (Z \cup Z')$

$= [(X \cap Y) \cup (X \cap Y')] \cap (Z \cup Z')$ ( Distributive Property )

$= (Z \cup Z') \cap [(X \cap Y) \cup (X \cap Y')]$ ( Commutative Property )

$= [(Z \cup Z') \cap (X \cap Y)] \cup [(Z \cup Z') \cap (X \cap Y')]$ (Distributive Property)

$= [(X \cap Y) \cap (Z \cup Z')] \cup [ (X \cap Y') \cap (Z \cup Z')]$ (Commutative Property)

$= [(X \cap Y \cap Z) \cup (X \cap Y \cap Z')] \cup [(X \cap Y' \cap Z) \cup (X \cap Y' \cap Z')]$ (Distributive Property)

$= (X \cap Y \cap Z) \cup [X \cap (Y-Z)] \cup [X \cap (Z-Y)] \cup [(X-Y)-Z]$ ( Set Difference , defined as $A-B=A \cap B'$ .)

Therefore,

$n(X)=n(X \cap Y \cap Z)+n[X \cap (Y-Z)]+n[X \cap (Z-Y)]+n[(X-Y)-Z]$

$n(X)=5+20+25+50=\boxed{100}$ .