Bxtterman

Thank you SOOOO MUCH!! This helped Enormously.  I get the problem 100% now, and I appreciate you spending the time to type all this out. I appreciate it!!

This response had something to do with computers and programming, matrices and vectors (all of which are too advanced for me.) I already saw it, however I did not get ANY of it, so I came here. :)

Do you know how you got this equation? Could you possibly break it down for me? Thank you so much for your answer, by the way. It is very elegant.

Thank you for your answer. My Approach initially was that I saw f(x) = inverse is the same as f(f(x)) = x, so I tried plugging it in, but that was more complex (though I think I may have gotten the correct answer that way too). I think that if I tried this method and thought about it in a more logical perspective, it would have been much better. Thank you!

How would you use an inverse matrix?

Great reminder at the bottom there.

Why are there 3 choices for C?

Thanks! I didn't think about using imaginary numbers, only rational haha.

The forum locks questions to prevent spam after they are posted for a certain period of time. This keeps users and the site itself safe and managable. Furthermore, people can use old post to farm experience, as was a problem with Reddit a while back, I remember when the site was down :)

Oh no @Melody moderator now. NOW! lol what is happening

Wait, is this AoPS week 24?!!!!!!!! I'm taking the same class as you I think!

How are they repeated?

yeah!

3) In the word REARRANGE we have 9 letters.

For these letters to be distinct, we must think of each letter like a gumball in a gumball machine. Every time I use one,  there of less to be used.

We also need casework for each length of word:

CASE 1: 1 letter words

9  one letter words.

CASE 2: 2 letter words

\(9\cdot8=72\) two letter words

CASE 3: 3 letter words.

\(9\cdot8\cdot7=504\)

CASE 4: 4 letter words

\(9\cdot8\cdot7\cdot6=3024\)

CASE 5: 5 letter words

\(9\cdot8\cdot7\cdot6\cdot5=15120\)

CASE 6: 6 letter words

\(9\cdot8\cdot7\cdot6\cdot5\cdot4=60480\)

CASE 7: 7 letter words (almost done!)

\(9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3=181440\)

CASE 8: 8 letter words

\(9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2=362880\)

CASE 9: (DONE!)

\(9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=362880\)

Now we add all the cases together for the total number of arrangments:

\(9+72+504+3024+15120+60480+181440+2(362880)=\boxed{986409}\)

SIDENOTE:

You can subsitute the decending numbers with a factorial

\(5\cdot4\cdot3\cdot2\cdot1=5!\)

\(5\cdot4\cdot3=\displaystyle\frac{5\cdot4\cdot3\cdot2\cdot1}{2\cdot1}=\displaystyle\frac{5!}{2!}\)

Just some rules for future reference, hope you enjoy and use them.

As always if you have any questions just ask.

Goodnight! 🌙

2) First we calculate how many possibilities before: 

\(9\cdot8\cdot7\cdot6\cdot5=15120\)

Now we calculate the possibilites now:

\(9\cdot9\cdot9\cdot9\cdot9=9^5=59049\)

We can get the difference:

\(59049-15120=\boxed{43929}\)

You can solve this problem easily by plotting it in on a coordinate plane and probability. 

NOTE BEFORE WE START. I MAKE MISTAKES AND I HAVE IN THE PAST PLEASE CHECK THIS ANSWER!

Thank you for reformatting.

I just graphed this. it's a diamond with equal side lengths.

We first see that there are 4 ways to get to the first letter, A

We add the first part to our equation: \(1\cdot4\)

From A, we can get to the next letter, T

We can get to there 2 ways from every A. 

Our equation is now: \(1\cdot4\cdot3\)

Finally, we can get to the letter H. 

For 4 of the 8 letter T's there are 2 H's.

Our equation is now \(1\cdot4\cdot3\cdot2=24\)

Next, we count the other 4. We have 3 H's for every T.

\(1\cdot4\cdot3\cdot3=36\)

We can add both cases:

\(24+36=\boxed{60}\)

Thank you man, I probabaly got this one wrong it's 1 AM and I'm answering math problems smh.

I wanna cuss but I can't haha but I need to vent everyday.

PLEASE PLEASE PLEASE PLEASE PLEASE PLEASE FORMAT PLEASE PLEASE

I'm like really pissed idek why this infuriates me. I have to give you props for effort in at least putting it in latex instead of others who just paste it in.

1) Spaces are removed in LaTeX

2) LaTeX in this forum is for EQUATIONS only

3) Noone will solve your question if you make it completly illegible, is shows little effort

4) This problem is totally OK with just text, just go to Notepad and CTRL+H and find and delete the H.

5) I like your username

So yeah. Reformat and me or someone will try to solve your question.

LOL! idk what the question was but it must have been really bad to p**s off melody. BTW-happy 100k points!