Melody

This question is actually very ambiguous.

Sometimes an outcome appears to be valid but if it is rotated it becames invalid.

Since rotations are considered euqivalent then that should mean that the original outcome is also invalid.

I tried to take this into acoount with my answer.  I have said both must be invalid.

It is the diagonal condition that causes this problem.

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Anyway guest, you have not attempted to explain how you got your answer. So it is impossible for anyone to consider your logic.

Thanks  answering Guest.

8 students are planning on splitting up into 3 (non-empty) study groups. How many ways can they split up into these study groups?

Note: It is possible that a study group contains only 1 person.

8 Students

If only the number of students in each group counts then

S*S*S*S*S*S*S*S       chose 3 stars from the 7   =  7C2  = 21 ways

(This is the same answer as guest got )

OR

If the exact students in each group is important then:

1,1,6        8C6               = 28

1,2,5         8C5*3           = 168

1,3,4         8C4*4           =  280

2,2,4         8C4*4C2      = 420

2,3,3         8C3*5C3      = 560

28+168+280+420+560 = 1456  ways

The first option looks right to me.  The second one I am much less sure about.

No guarentees for either.

Asker guest:  If you have a number answer can you please share it with the rest of us.

(If you believe it to be wrong please say why you think so.)

\(\boxed{2nd}\quad \boxed{atan}\quad \text{your number }=\)

arctan is the same as inverse tan.

ALSO

How many ways are there to put 9 differently colored beads on a grid if the purple bead and the green bead cannot be adjacent (either horizontally, vertically, or diagonally), and rotations and reflections of the grid are considered the same?

A grid of 3 dots by 3 dots.      9 colours but pink and green cannot be next to each other.

I really do not know but here is my 'guess' 

So there are 7 colours that can go into the middle.

Rotations are the same just put a green in a corner

You cannot have a pink next to it but you cannot have a pink on the adjacent corners either because it it is rotated they will become diagonally adjacent,

So

I get   7*1*6*5*4*3 *3*2*1 = 15120

There are 4 axes of symmetry so divide by 2^4=16

15120/16 = 945

I doubt that it is correct though.

If you know what the number answer is then please share that knowledge with the forum.

I assume you do not know how to do it but do you know what the answer is? 

Or do you have any information that could help answerers.

Please answer either way.   

I did it in a circle and I realize now that this is not right. You want it in a grid.

But you have not said what shape the grid is. I suppose it is supposed to be a 3 by 3 grid but this has not been stated.

So I might think about it again......

Lots try counting them systematically

1 x 2 ?

1 x 3 ?

1 x 4 ?

1 x 5 ?

2 x 3 ?

2 x 4 ?

2 x 5 ?

3 x 4 ?

3 x 5 ?

4 x 5 ?

that is it.

Please no one else interfer.  Asker guest can do this by themselves now.  How else will they learn?