A Fano\:table is a table with three columns where
- each entry is an integer taken from the list 1,2,3,\dots,n, and
- each row contains three different integers, and
- for each possible pair of distinct integers from the list 1,2,3,\dots,n, there is exactly one row that contains both of these integers.
The number of rows in the table will depend on the value of n. For example, the table shown is a Fano table with n=7. (Notice that 2 and 6 appear in the same row only once, as does every other possible pair of the numbers 1,2,3,4,5,6,7.) For how many values of n with 3\le n\le12 can a Fano table be created?
Answer Choices
A. 2
B. 3
C. 5
D. 6
E. 7