Finding the number of elements of a set 
Let $S$ be the set of all integers from $100$ to $999$ which are neither divisible by $3$ nor divisible by $5$. The number of elements in $S$ is

*

*$480$

*$420$

*$360$

*$240$

My answer is coming out as $420$, but in the actual answer-sheet the answer is given as $480$. Why is my answer not correct? Please, someone help.
 A: From set theory, $|A \cup B|$, the number of elements in the set $A \cup B$, can be expressed as
$$|A \cup B| = |A| + |B| - |A \cap B|$$
You want to compute $$|S| - |A \cup B|$$
Where
\begin{align}
   S &= \{100 \dots999\} \; \text{and} \; |S| = 900. \\
   A &= \{102, 105, 108, ..., 999\} \; \text{and} \; |A| = 300. \\
   B &= \{100, 105, 110, ..., 995\} \; \text{and} \; |B| = 180. \\
   A \cap B &= \{105, 120, ..., 990\} \; \text{and} \; |A \cap B| = 60. \\
   \hline
   |A \cup B| &= 300+180 - 60 = 420.\\
   |S| - |A \cup B| &= 900 - 420 =480.
\end{align}
A: $900\times\frac{2}{3}\times\frac{4}{5} = 480$
Reference Inclusion–exclusion principle

Sorry, I mangled the formula and rightly picked up a down-vote.  I fooled myself because it gave the right answer for 3 and 5.  I am correcting it.  Please see Steven Gregory's answer for the proper inclusion–exclusion treatment.
A: Solve[{100 <= n <= 999, ! Element[n/3, Integers], ! 
    Element[n/5, Integers]}, n, Integers] // Length

(*  480  *)

A: Length@Complement[Range[100, 999],Union[Range[102, 999, 3], Range[100, 999,5]]]

(*480*)
