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Код

 public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();

            BindingList<MyClass> bList= new BindingList<MyClass>();
            
            bList.Add(new MyClass("Ivan", 23));
            bList.Add(new MyClass("Igor", 63));
            bList.Add(new MyClass("Anna", 13));
            bList.Add(new MyClass("Mike", 41));
            this.listBox1.DataSource = bList;

            SortBindingList(bList);
        }


        //реализация сортировки для BindingList
        private void SortBindingList(BindingList<MyClass> bList)
        {
            BindingList<MyClass> bList2 = new BindingList<MyClass>();
            IEnumerable<MyClass> e = bList.OrderBy(man => man.name);

            bList2.Clear();
            foreach (MyClass my in e)
            {
                bList2.Add(my);
                bList.Remove(my);
            }

            foreach (MyClass my in bList2)
            {
                bList.Add(my);
            }
        }
    }

    //Пример класса для листбокса
    class MyClass
    {
        public string name;
        public int age;
        public MyClass(string name, int age)
        {
            this.age = age;
            this.name = name;
        }
        public override string ToString()
        {
            return this.name + "\t" + this.age;
        }
    }