Counting total number of matches with grep instead of just how many lines match

try this:

grep -o -E "your expression" file |wc -l

well, -E is just an example, it could be -P, -F etc. point is -o

test:

kent$  echo "abc xxx yyy"|grep -cP "[a-z]{3}"      
1

kent$  echo "abc xxx yyy"|grep -oP "[a-z]{3}"|wc -l
3

There is a -o flag which indicates that only the matched subsection of the line should get printed.

Use that in conjunction with wc -l:

grep -o "part of line" | wc -l

man grep explains it as well.