How to show the next button when 3/4 flipcards are clicked in articulate storyline?

Because the flip cards can be toggled, you can't rely on all of them to be in the Selected state at the same time. So I suggest you use T/F variables to track whether each card has been clicked. 

• For each card, create a T/F variable with the default value of False.
• Add a trigger to each card that adjusts the corresponding variable to True when the card is clicked. 
• Add a trigger to each card that changes the Next button to Normal with conditions that each variable = True.

Here's more info: 

• TIP: Controlling the NEXT Button 101 | Articulate - Community
• The Value of Variables | Articulate - Community
• Storyline 360: Working with Triggers | Articulate - Community