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The Chicago Bears score 18 points in a football game. In how many different ways can the Bears score these points? Points are scored as follows: a safety is 2 points, a field goal is 3 points, a touchdown is 6 points, and a point after touchdown (PAT) is 1 point (a PAT can't be scored unless a touchdown is scored first).

My Solution

From here on out, I abbreviated Touchdown as TD, Field Goal as FG, and point after touchdown as PAT. I created a systematic list, beginning with the maximum number of touchdowns scored and then the maximum amount of field goals with 2 touchdowns and so on until I was left with no further solutions. To make it easier for you to read, I just inputted my data into Excel and took a screenshot.

I am wondering if I accidently skipped over a possible way when making my list. Keep in mind, there are NO 2-point conversions after a TD is scored. I also assumed that you could miss a PAT, even though it is unlikely (but, so are some of these solutions below - 9 safeties, really?).

My systematic list:

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I got a total of 14 ways.

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    Ah, valid point Michael. It did not seem too problematic when i was working through the problem (trying to figure out numbers that add up to x when y amount of TDs, FGs, etc.). However, I will note your strategy may be easier for readers to follow and for me to perform even - thanks.2012-02-14

1 Answers 1

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A really simple and elegant solution is a polynomial expansion. You can consider each way a team could score (touchdowns with 2-pt conversion, touchdown with xp, touchdown, field goal, etc.) and the number of times it is possible for a team to have scored in those ways. In your example, a team could score at most 2 touchdowns with extra points, 6 field goals, or 9 safeties. This can be represented as:

$(1+x^7+x^{14})(1+x^6+x^{12}+x^{18})(1+x^3+x^6+x^9+...+x^{18})$$(1+x^2+x^4+x^6+...+x^{18}).$

The exponents denote the number of points scored, while the coefficients denote the number of possible ways to score that many points. We begin each polynomial with 1 because there is one way to score 0 points with each type of scoring play. This expansion yields 14x^18, showing us that there are exactly 14 combinations of scoring plays (excluding 2-pt conversions) that could result in 18 points for the Monsters of the Midway.