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Dancing (Posted on 2003-02-16) Difficulty: 2 of 5
In a popular dancing game there are 4 directions: up, down, left, right. In the game these arrows may appear once every quarter-note or eight-note. Also two arrows may appear at the same time in the same beat. Three or four arrows cannot.

In a song that is 1 minute long and plays at 1 bar(4 quarter notes) every 4 seconds, what are the total possible number of different dances that can exist.

See The Solution Submitted by Alan    
Rating: 4.3333 (3 votes)

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Solution Solution | Comment 4 of 6 |

More than 23040. (actual solution at the end) Let's see.

If I understood this correctly, it's not a simple permutation. I assumed that an up arrow and a right arrow in the same beat could be considered different than an up arrow then a right arrow. Therefore there are 4x3=12 ways to choose two, but you could also just choose 1, so there are really 16 ways to choose arrows for the beats.

I also assumed that by "per beat" you meant "per note," NOT two if it's an eighth note and one if it's a quarter. If I was wrong about that, well, I blame you. Hahaha.

So when you look at it, you have to figure out how many notes you're dealing with per bar and multiply that times 16. Well, a bar could be all quarter notes (1way@4notes), 3 quarter notes and one set of eighth notes, but wait. A set of eighth notes need not appear as a set. One eighth note could appear at the beginning, the other at the end. Yikes. It's a combination thing. For 3 Quarter and 2Eighth there are 5 choose 2 combo (or 5 choose 3 combo, the answer is the same). So there are 5!/3!*2!=10. For 2Quarter and 4Eighth there are 6!/4!*2!=15. For 1QUarter and 6 Eighth, there are 7 choose 1, or 7. And there's only one way to have 8 eighths. SO there is a total of 34 different ways for the notes to go per bar. BUT you can't simply multiply can you. The number of notes now matters, SO you multiply the number of ways times the number of notes times the 16 arrow permutations per note...

4Q 1*4*16=64
3Q2E 10*5*16=800
2Q4E 15*6*16=1440
1Q6E 7*7*16=784
8E 1*8*16=128

For a total of 3216 dance arrangements per bar. There are 15 bars per minute, so 48,240. Wow that was a good one. Did I actually get it right?
  Posted by Lawrence on 2003-08-27 01:39:27
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