When I was writing The Killing I knew that structure was going to be important. This was a massive work and it had to be divided into parts, days and individual scenes. But I wanted to keep an eye on the length of those parts.
Simple solution: count the number of scenes in each part. And yes, Scrivener can do this for you like this.
But you won’t get that information out of the box. Go into Preferences, Appearance and choose Binder: Shows subdocument counts.
Now you’ll get the counts you see in the first screenshot. Note that The Killing has 526 scenes in all (including the short epigraph). The manuscript count at the top does not include anything outside the manuscript such as character, places and research information.
PS. I keep my sub documents in folders. But if you make your documents sub to another document I think this works too… though I don’t so I haven’t checked.
Update. Do please read the important caveat added by Sean C. in the comments below.