Now suppose that letter 9 was not typed before lunch. It may already be waiting on top of the pile when the secretary returns from lunch, or it may be added at any time during the afternoon. Apart from letter 9, there can again be any subset of letters 1 to 7 remaining to by typed. If there are k such letters, then letter 9 can be inserted at any of k+1 points (before all of them, after all of them, or at any intermediate point). The total number of orders is thus .
So the final answer is .