This one caught my interest so I wrote a console app to do it.
Andomar answered a few moments before I did (+1) from me.
This is my independent solution but very close to his and no simpler...
class Program {
    private class CustomerDto {
        public int CustomerId { get; set; }
        public string CustomerName { get; set; }
    }
    private class ItemDto {
        public int ItemId { get; set; }
        public string ItemName { get; set; }
    }
    private class OrderDto {
        public int Id { get; set; }
        public int ItemId { get; set; }
        public int CustomerId { get; set; }
        public int Quantity { get; set; }
    }
    private class CustomerOrderDto {
        public int CustomerId { get; set; }
        public int ItemId { get; set; }
        public int TotalQuantity { get; set; }
    }
    static void Main(string[] args) {
        List<CustomerDto> Customers = new List<CustomerDto>() { 
            new CustomerDto() { CustomerId = 1, CustomerName = "one"},
            new CustomerDto() { CustomerId = 2, CustomerName = "two"},
            new CustomerDto() { CustomerId = 3, CustomerName = "three"}
        };
        List<ItemDto> Items = new List<ItemDto>() { 
            new ItemDto() { ItemId = 1, ItemName = "item one"},
            new ItemDto() { ItemId = 2, ItemName = "item two"},
            new ItemDto() { ItemId = 3, ItemName = "item three"}
        };
        // customer1 has 2 orders for item 1, 0 for item 2 or 3
        // customer2 has 1 order for item 2, 0 for item 1 or 3
        // customer3 has 1 order for item 2, 1 order for item 3 and 0 for item 1
        List<OrderDto> Orders = new List<OrderDto>() { 
            new OrderDto() { Id = 1, CustomerId = 1, ItemId = 1, Quantity = 3 },
            new OrderDto() { Id = 1, CustomerId = 1, ItemId = 1, Quantity = 5 },
            new OrderDto() { Id = 1, CustomerId = 3, ItemId = 2, Quantity = 5 },
            new OrderDto() { Id = 1, CustomerId = 3, ItemId = 3, Quantity = 5 },
            new OrderDto() { Id = 1, CustomerId = 2, ItemId = 2, Quantity = 5 }
        };
        List<CustomerOrderDto> results = (from c in Customers
                                          from i in Items
                                          join o in Orders on
                                            new { c.CustomerId, i.ItemId } equals
                                            new { o.CustomerId, o.ItemId } into oj
                                          from o in oj.DefaultIfEmpty()
                                          let x = o ?? new OrderDto() { CustomerId = c.CustomerId, ItemId = i.ItemId, Quantity = 0 }
                                          group x by new { x.CustomerId, x.ItemId } into g
                                          select new CustomerOrderDto() {
                                              CustomerId = g.Key.CustomerId,
                                              ItemId = g.Key.ItemId,
                                              TotalQuantity = g.Select(x => x.Quantity).Sum()
                                          }
                                          ).ToList();
        foreach (var result in results) {
            Console.WriteLine("Customer {0} purchased {1} units of item {2}",
                result.CustomerId, result.TotalQuantity, result.ItemId);
        }
        Console.ReadKey(true);
    }
}