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! 🌙