Bowling Counting algorithm from InterviewStreet

Question

I don't know if this is the right section... but here goes:

Last weeks contest on interviewstreet (Code Sprint 3) had a problem called bowling. (10 pin bowling, N frames). The point is to count the number of ways to score M points by playing N frames.

Problem Statement is here: http://pastebin.com/cyeLML8U

I'm pretty sure I've solved the problem using 2 dimensional DP. However, I get the 3rd sample data wrong (1 Frame, 25 points). The sample answer is 1, however I get 6.

This is their explanation of the sample answer:

For the third case, there is only 1 way. Score a strike in the first frame, score another strike with the first extra ball, and an additional 5 with the second extra ball.

However, can't you score a strike in the first (and only) frame, then score any of the following in the subsequent extra frames?

10 5
9 6
8 7
7 8
6 9
5 10

I can't wrap my head around why "1" is the right answer.... I've looked on wikipedia for the rules too.

Their answer is probably right, and I'm probably overlooking something REALLY obvious. Can anyone tell me what's wrong with my answer?

Answer 1

You cannot get 9 pins with the first extra ball and then 6 pins with the second extra ball because there is only 1 pin left standing when you bowl the second extra ball.

Answer 2

But if you don't get a strike on the second ball, you only have the opportunity to "pick up the spare." That is, you only get 10 pins. So if you get a strike on the first ball and then 9 pins on the second ball, the most you can get on the third ball is 1.