This depends on the density of the specific aircraft seat capacity (even at the same airline they may have different density with some configured with J/Y and others configured F/J/Y); length of flight; level of service; and cabin crew contracts.

At a minimum, the US FAA requires one F/A per 50 seats. On some US versions of the 777-200 that would have been a minimum crew of 6(like the DC-10). However, I believe in each carriers Evacuation Certification tests that FAA minimum crew ended up being 8. Most US carriers staff the 777-200 with 8 on domestic routes (unless operated in trans-con markets, with higher service levels) and 11 or 12 internationally, depending on passenger load and service elements on the route.

That said, you should expect the 777-300 to have more cabin crew than the 777-200 as it is a bigger plan, with more emergency exits, and higher seating capacity.