// ObservableObject is a custom base class that implements INotifyPropertyChanged
internal class MainWindowVM : ObservableObject
{
    private ObservableCollection<Item> mItems;
    private HashSet<Item> mCheckedItems;

    public IEnumerable<Item> Items { get { return mItems; } }

    public string Text
    {
        get { return _text; }
        set { Set(ref _text, value); }
    }
    private string _text;

    public MainWindowVM()
    {
        mItems = new ObservableCollection<Item>();
        mCheckedItems = new HashSet<Item>();
        mItems.CollectionChanged += Items_CollectionChanged;

        // Adding test data
        for (int i = 0; i < 10; ++i)
        {
            mItems.Add(new Item(string.Format("Item {0}", i.ToString("00"))));
        }
    }

    private void Items_CollectionChanged(object sender, NotifyCollectionChangedEventArgs e)
    {
        if (e.OldItems != null)
        {
            foreach (Item item in e.OldItems)
            {
                item.PropertyChanged -= Item_PropertyChanged;
                mCheckedItems.Remove(item);
            }
        }
        if (e.NewItems != null)
        {
            foreach (Item item in e.NewItems)
            {
                item.PropertyChanged += Item_PropertyChanged;
                if (item.IsChecked) mCheckedItems.Add(item);
            }
        }
        UpdateText();
    }

    private void Item_PropertyChanged(object sender, PropertyChangedEventArgs e)
    {
        if (e.PropertyName == "IsChecked")
        {
            Item item = (Item)sender;
            if (item.IsChecked)
            {
                mCheckedItems.Add(item);
            }
            else
            {
                mCheckedItems.Remove(item);
            }
            UpdateText();
        }
    }

    private void UpdateText()
    {
        switch (mCheckedItems.Count)
        {
            case 0:
                Text = "<none>";
                break;
            case 1:
                Text = mCheckedItems.First().Name;
                break;
            default:
                Text = "<multiple>";
                break;
        }
    }
}

// Test item class
// Test item class
internal class Item : ObservableObject
{
    public string Name { get; private set; }

    public bool IsChecked
    {
        get { return _isChecked; }
        set { Set(ref _isChecked, value); }
    }
    private bool _isChecked;

    public Item(string name)
    {
        Name = name;
    }

    public override string ToString()
    {
        return Name;
    }
}