Could you please help me figure out how I could bind "SelectedItems" of a listbox?

Here I have a 'listbox_2' that contains a list of fruits and vegetables
There's another 'listbox_1' that contains groups of fruits and vegetables (under a list<of string> format...maybe is it my first error? Should it rather be a list of fruits/vegetables?) and a recipe.

I would like that on each group selection (in listbox_1), the corresponding fruits and vegetables would be selected (in listbox_2)...and make it possible to modify that list
Unfortunately, the way to achieve this behavior stays obscure to me...could you please help?

Here's what I've set up untill now:

public partial class MainWindow : Window
        ItemList il;
        GroupList gl;

        public MainWindow()
        private void Window_Loaded(object sender, RoutedEventArgs e)
            il = new ItemList();
            ICollectionView cvs = CollectionViewSource.GetDefaultView(il);
            cvs.SortDescriptions.Add(new SortDescription("_type", ListSortDirection.Ascending));
            cvs.SortDescriptions.Add(new SortDescription("_name", ListSortDirection.Ascending));
            cvs.GroupDescriptions.Add(new PropertyGroupDescription("_type"));
            ListBox2.ItemsSource = cvs;

            gl = new GroupList();
            ICollectionView cvt = CollectionViewSource.GetDefaultView(gl);
            ListBox1.ItemsSource = cvt;

    public class Item
        public string _type { get; set; }
        public string _name { get; set; }
        public Item()
    public class ItemList : ObservableCollection<Item> {
        public ItemList() {
            base.Add(new Item() { _type = "fruit", _name = "apple" });
            base.Add(new Item() { _type = "vegetable", _name = "potato" });
            base.Add(new Item() { _type = "fruit", _name = "banana" });
            base.Add(new Item() { _type = "vegetable", _name = "tomato" });
            base.Add(new Item() { _type = "fruit", _name = "pear" });
            base.Add(new Item() { _type = "vegetable", _name = "salad" });
            base.Add(new Item() { _type = "fruit", _name = "orange" });
            base.Add(new Item() { _type = "vegetable", _name = "onion" }); 

    public class Group
        public string _groupname { get; set; }
        public List<String> _members { get; set; }
        public string _recipe { get; set; }
        public Group()
    public class GroupList : ObservableCollection<Group>
        public GroupList()
            base.Add(new Group() { _groupname = "Group_1", _members = new List<String>() { "apple", "salad" }, _recipe = "Do this and do that" });
            base.Add(new Group() { _groupname = "Group_2", _members = new List<String>() { "banana", "onion" }, _recipe = "Don't do that and do this" });
<Window x:Class="WpfApplication1.MainWindow"
        Title="MainWindow" Height="350" Width="525"
        <Label Margin="12,0,378,283" Content="Group"></Label>
        <Label Margin="190,0,200,283" Content="Members"></Label>
        <Label Margin="309,0,81,283" Content="Recipe"></Label>
        <TextBox Margin="309,34,12,12" DataContext="{Binding SelectedItem, ElementName=ListBox1}" Text="{Binding Path=_recipe, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}"></TextBox>
        <ListBox Margin="12,34,378,12" Name="ListBox1" SelectionMode="Single">
                    <TextBlock Text="{Binding _groupname}" />
        <ListBox Margin="190,34,199,12" Name="ListBox2" SelectionMode="Multiple" SelectedValuePath="_name">
                    <TextBlock Text="{Binding _name}" />
                        <Style TargetType="{x:Type GroupItem}">
                            <Setter Property="Template">
                                        <Expander Header="{Binding Name}" IsExpanded="True">
                                            <ItemsPresenter />
EDIT: from my reading, it seams that listboxes couldn't 'easily' bind 'SelectedItems'..maybe then a solution to bind to a checkbox component inside the listbox item?

Thanks for any kind help!