Count results in Views using aggregation
Yes, it is possible in Views 3 out of the box.
The idea is the same as in SQL aggregations. Lets see an example:  


Edit your view and enable Views aggregations:

Remove default sort criteria. 
Add fields “Content: Type” and “Content: Nid”:

Select COUNT function for Content: Nid:



To see what happens just turn on checkbox “Show the SQL query” at global Views settings page.  

SELECT node.type AS node_type, COUNT(node.nid) AS nid
FROM 
{node} node
WHERE (( (node.status = '1') ))
GROUP BY node_type
LIMIT 10 OFFSET 0


So, we are grouping nodes by node_type and calculating count for this groups.