One of my heroes is an anonymous primate who maintains a site called balmoralsoftware.com. If you click on that link, you will find an unassuming webpage with a number of links in small print. The last of these links reads "Bowling Scores". If you click on that link, you will be transported to a wondrous world of the most detailed mathematical treatment of bowling scores ever written. Whenever I do so, I feel like Charlie entering Willy Wonka's chocolate factory. Yes, I have a problem. If you scroll down on that page, you will find a list that tells you how many different ways there are to arrive at any particular bowling score. For example, you will find that there are 2,447,444,695,948,898 pathways to a score of 124. That means that are more than 2.4 quadrillion ways to score 124.
These numbers become much more tractable as you get into the high 200's. A major break occurs between the scores 290 and 291. According to this wonderland of bowling scores, there is only one way to arrive at a score of 291. This is accomplished with eleven consecutive strikes followed by a one on the bonus ball in the 10th. This makes me wonder if anybody has ever actually scored 291, but my question concerns 290. According to "Mr. Balmoral," there are eleven ways to score 290. The first should be obvious... eleven straight strikes followed by a gutter ball. Here is my question:
What are the other ten ways to score 290?
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