IIUC, you could use value_counts()
for that:
df[['Last Name','First Name']].value_counts().size
3
For another example, if you start with this extended data frame that contains some dups:
Last Name First Name
0 Smith Bill
1 Johnson Bill
2 Smith John
3 Curtis Tony
4 Taylor Elizabeth
5 Smith Bill
6 Johnson Bill
7 Smith Bill
Then value_counts()
gives you the counts by unique composite last-first name:
df[['Last Name','First Name']].value_counts()
Last Name First Name
Smith Bill 3
Johnson Bill 2
Curtis Tony 1
Smith John 1
Taylor Elizabeth 1
Then the length of that frame will give you the number of unique composite last-first names:
df[['Last Name','First Name']].value_counts().size
5