CPhill

"At most" connotes 1, 2 or 3 groups

For one group = one way....all of them together

For two groups.....2 + 2  or 3 + 1  = two ways   ...{the oranges are indistinguishable, so 1 + 3 is the same grouping as 3 + 1 }

For 3 groups = 1 + 1 + 2  = one way   {again, they are indistinguishable}

So ..... four ways.....

C(15,2) * C(10,3) = 12,600 ways

Well...if I can use my secret decoder ring, I think you're saying that for every 2 votes Billy got, Haley got 7.

So...the ratio of votes for Haley to Billy = 7 : 2

Here's the justification for my answer....

Consider a "birdseye" view of the table....

One person either occupies the left seat on one side of the table or the right seat on that side.

And for each of these arrangements, the other 7 seats can be filled in all possible ways = 7! ways.

So, 2 x 7!   = 10,080 ways

"Rotations" of the table don't matter......one rotation would look like any other for a particular arrangement.

(2x^3)(4x^1/3) =

8x^3 * x^(1/3) =

(8x^3)³√x

One person can be 'anchored" in one of two seats on one side.......the other seven people can be arranged in 7! ways.....so....

2 x 7!  =

2 x 5040 = 10,080 ways

Consider that the three are anchored as a "block' .....  they can be seated in 3! = 6 ways

And the other 5 people can be arranged  in 5! =120 ways

So

6 x 120  = 720 different arrangements

(9+3)-3x+34^2-8(2-9)=38x    simplify

12 - 3x + 34^2 - 8(-7) = 38x

12 - 3x +  1156 + 56 = 38x

1223 - 3x = 38x      add 3 x to both sides

1224 = 41x        divide both sides by 41

x =  about 29.8536585365853659