WPF DataGridComboBoxColumnのItemsSourceにコレクションをバインドする方法

たぶんよろしくはない。
改善策があれば教えていただきたいです。
(enumerationであえてバインドしていません)

Person.cs

    class Person
    {
        public string ID { get; set; }
        public string Name { get; set; }
        public string BirthPlace { get; set; }
    }

MainWindow.xaml.cs

    public partial class MainWindow : Window
    {
        private List<string> birthPlaces;
        public MainWindow()
        {
            InitializeComponent();

            birthPlaces = new List<string>
            {
                "Tokyo",
                "Osaka",
                "Aichi"
            };

            ObservableCollection<Person> persons = new ObservableCollection<Person>
            {
                new Person { Name="Aoki", ID="0", BirthPlace="Tokyo"},
                new Person { Name="Aikawa", ID="1", BirthPlace="Osaka" },
                new Person { Name="Adachi", ID="2", BirthPlace="Aichi" }
            };
            dgcbc.ItemsSource = birthPlaces;
            DataContext = persons;
        }
    }

MainWindow.xaml

<Window x:Class="DataGridComboBoxColumnBindingSample.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        xmlns:local="clr-namespace:DataGridComboBoxColumnBindingSample"
        mc:Ignorable="d"
        Title="MainWindow" Height="350" Width="525">
    <Grid>
        <DataGrid ItemsSource="{Binding}" AutoGenerateColumns="False">
            <DataGrid.Columns>
                <DataGridTextColumn Header="Name" Binding="{Binding Name}"/>
                <DataGridComboBoxColumn x:Name="dgcbc" SelectedItemBinding="{Binding BirthPlace}"/>
            </DataGrid.Columns>
        </DataGrid>
    </Grid>
</Window>