

Does the checkbox of the first row mean it's checked or not??
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It means some options of the subtree of options of that option are selected or that the box is checked for some of the objects. E.g. when using a single properties dialog for multiple files. If you select files Sometimes an option tree is used like the one below.
So if all options below "My meal" are checked, so will "My meal". But if only one is not (doesn't matter if it's "Sandwich" as a whole or just a sub option of "Sandwich"), "My meal" will be like on your screen shot. | |||||||||||||||
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Programmatically speaking, There are 3 states to a Check Box - Checked, Unchecked and Indeterminate.
Example - http://msdn.microsoft.com/en-us/library/ms743611(v=vs.85).aspx The following graphics illustrate the three states of a CheckBox. The graphics show a dialog that shows the properties of files in an application called accesstext. The attributes of the files are indicated with a CheckBox. In the first graphic the file is Read-only (CheckBox is checked) and the second is Read\Write (CheckBox is unchecked). In the first two graphics the information concerns only one file Pane1 but the third graphic concerns all the files in the accesstext application. Some of the files are Read-only and some are Read\Write the attributes for the entire application cannot be determined (CheckBox is null). Checked
Unchecked
Indeterminate
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The checkbox has three states:
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Your comments indicate that you're looking at folders in Explorer, so here is the obligatory Old New Thing link to answer the question you really have: Why is the readonly property for folders so strange? What it boils down to is that the read-only and system flags tell Explorer to check the folder for a desktop.ini file that specifies folder customisations. Most programs ignore the read-only flag for folders, at least in the traditional sense of "don't allow anything to write to this folder." That's why you see the indeterminate status in Explorer. There's also MS KB Article 326549:
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