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.

Sorry, didn't mean to post the last empty post.
Please don't dog me if I am on the wrong track here

Since there are 4 seconds per bar there is 1 second per quarter note

I have figured this with rests and without

each second could contain one of these events

1 quarter rest*
1 quarter arrow
2 quarter arrows
1 eighth rest and 1 eighth rest*
1 eighth rest and 1 eighth arrow*
1 eighth rest and 2 eighth arrows*
1 eighth arrow and 1 eight rest*
1 eighth arrow 1 eighth arrow
1 eighth arrow 2 eighth arrows
2 eighth arrows 1 eighth rest*
2 eighth arrows 1 eighth arrow
2 eighth arrows 2 eighth arrows

*contains a rest

There are 12 possible events here per second
there are 60 seconds in the dance so there could be 12^60 dances possible

if we eliminate the possibility for rests (which were not mentioned in the problem) then we can eliminate the events that contain them.

that leaves 6 possible events for each second 6^60 possible dances.

So 12^60 if rests are allowed (5.6347514353166785389812313795981e+64)
and 6^60 without rests(4.8873677980689257489322752273775e+46)

Posted by Ron on 2003-02-19 18:05:01