The question of how many CCs is ideal has been discussed a lot in this forum. I wanted to present some data from recent experience. Everyone's mileage may vary, but more data is always useful.
Basics: I have only CCs as open accounts. My AAoA is 7 years. I have had 1 to 4 CCs on EQ and TU (and the two CRs don't have identical data).
TU: Going from 1 to 2 CCs increased my score by at least 20 points. I say at least 20 because the net increase was 20. Presumably, I took a hit for "New account" (that I estimate to be about 5 points). The corresponding inquiry was not on TU and AAoA remained unaffected. A few months back, going from no CCs to 1 CC (that I've had for 10+ years but briefly stopped reporting because of some technical problems) increased my score by 40+ points.
EF: I've always had at least 2 CCs on EF. I really don't understand EF's FICO scoring. So I am just going to present as much of the info I think is relevant. The notation below is:
(Number of CCs with balance/Total CCs, Time since most recent account opened, Number of inquiries, AAoA)
Let's call the starting score 0. It's only been downhill from there. 
The symbol `*' in the third field means no accounts opened in the last two years.
Sep: (1/3, *, 1, 7): 0
Oct: (2/3, *, 1, 8): -16
Oct: (2/4, 1, 2, 7): -19
Nov: (1/4, 2, 2, 7): -11
Nov: (1/3, 2, 2, 7): -11
Nov: (2/3, 2, 2, 7): -19
Summary:
Going from 4 to 3 CCs made absolutely no difference, all else being identical in EF. So I suppose that going from 3 to 4 CCs does not help mix at all. From the TU data, it is clear that going from 0 to 1 CCs helped a lot (~40 points) and going from 1 to 2 CCs also helped quite a bit (20+ points). I don't know if or how much going from 2 to 3 CCs helps.
Utilization is somewhat of a mystery to me. Going from 1/3 to 2/3 seems to have impacted the score by -16 points in Oct, but going back to 1/3 brought back only 5 of those points (to -11) in Nov. So the remaining 11 points is likely attributable to the new account recency and inquiries. An oddity is that the new account recency and inquiries seem to have hurt by only 3 points in Oct (from -16 to -19). The only explanation I can think of is that AAoA changing from 7 to 8 years caused a rebucketing that caused the score to *drop* by 8 points and, in this new bucket, the new account recency and inquiries hurt by only 3 points. In the 7 year AAoA bucket, the new account recency and inquiries hurt by 11 points. In both buckets, going from 1/3 to 2/3 utilization hurt by 8 points.