1) For any , let us define
These above mentioned are the three combinatorial lines in . There is only one combinatorial line of type C , which is C itself. |A| = 4, there are four combinatorial lines , ,t in A and there are four combinatorial lines in .These are all the combinatorial lines in . Hence there are a total of 9 combinatorial lines in .
2) Consider any Subset . For every , fix any in A.
Then we get a combinatorial line defined as:
We further note that reach combinatorial line is of the form , since there is only one free coordinate in a combinatorial line.
Now , for any , the number of subsets of of size F is equal to
Further , for any subset , the number of choices for the remaining coordinates
is equal to
Hence the total number of combinatorial is equal to :
By Binomial Theroem .