Melody

Actually, you are right.   I don't know that.  I can see it is close to that but it may not be exact ....

Thanks for picking that up.   I 'll think about it a bit more ...

Let's include a slightly more helpful version of your graph here.

I haven't done it yet and I always have to look up the 'rules' when I am dealing with ellipses

but it is pretty clear that the focii are (-1,0) and (1,0)

I'd take it from there.  I might do it later.

idk Ginger.

It all looks good but I m not sure you used the right ambiguity setting.

If you miscalculated that then your whole answer is in dire jeopardy    

ok but without a pic it really does not make sense, not to me anyway.  

I expect there is supposed to be a pic with this question.

Seat Jill first.  Rotations are normally classed as the same so only one way.

Seat the 2 boys   2 ways

Seat the other 2 in the remaining seats   2 ways.

2*2=4 ways.

Quote from Google

"If a number is being divided by 9, add each of the digits to each other until you are left with one number (e.g., 1164 becomes 12 which in turn becomes 3), which is the remainder."

n (mod9) = S(n) (mod 9) = S(S(n)) (mod 9)     for all integer n

so

n = S(n) + S(S(n)) (mod  9)

becomes

n (mod9) = n (mod9) +n (mod9) 

2n=n (mod9)

2n=n+9k 

n=9k

n must be a muliple of 9

How many multiples of 9 are there between 100 and 999?

111-10  =  101  

If anyone thinks I have made a logic error please let me know, with your reasons of course.

...given what?

26 letters, 5 vowels

5 choices for the second letter, then 25 choices for the 3rd letter, then 26 choices for the first letter.

Now what?  

Have you graphed this with desmos?

I suggest you do so and attach a link to the graph.  It is always a good starting point.