Total number of pure 3BLD scrambles

[abunickabhi]

I want to know how many pure scrambles are there, with no cycle break or flipped piece in corners and edges. Also not more than 1 solved piece in edge and corner.

Or in other words, how many 3x3 states have a pure 11-cycle of edges and pure 7-cycle of corners?

 

[Exmestify]

from which orientaion?

 

[Fred]

Here's my attempt:
First there are 8 choices for solved corner and 12 for solved edge. Then a 7-cycle can be written as (1 4 6 2 7 3 5), for example. That means piece 1 goes to position 4, piece 4 goes to position 6, etc, piece 5 goes to position 1. In this representation, 1 is always at the start, and you have 6! ways to choose the rest of the cycle. Similarly there are 10! 11-cycles. Finally, for orientation there are 3^6 corner orientations and 2^10 edge orientations, since the solved piece has to be correctly oriented. So the answer is 8*12*10!*6!*3^6*2^10 = 187 quadrillion, or about 0.435% of all possible scrambles.

 

[bldcuber314]

from which orientaion?

Don't think it makes a difference

 

[tseitsei]

Here's my attempt:
First there are 8 choices for solved corner and 12 for solved edge. Then a 7-cycle can be written as (1 4 6 2 7 3 5), for example. That means piece 1 goes to position 4, piece 4 goes to position 6, etc, piece 5 goes to position 1. In this representation, 1 is always at the start, and you have 6! ways to choose the rest of the cycle. Similarly there are 10! 11-cycles. Finally, for orientation there are 3^6 corner orientations and 2^10 edge orientations, since the solved piece has to be correctly oriented. So the answer is 8*12*10!*6!*3^6*2^10 = 187 quadrillion, or about 0.435% of all possible scrambles.

Seems correct to me. That is how I thought about it too.

 

[abunickabhi]

Don't think it makes a difference

Yes it does not make a difference.

Any orientation is fine yo.