You cannot precompute this based just on the number of tokens. Bigrams are pairs of tokens which occur side-by-side (it is a term from n-gram models, where you have a notion of sequence). So in order to compute number of bigrams you have to slide a 2-token window through your data and check how many different pairs you find.
If you have N tokens coming from some data X, you can only say, that number of bigrams B is bounded as follows: N <= B <= N^2
, but the exact number requires the procedure outlined above.