Find the number of ways in which letters of the word ARRANGEMENT can be arranged so that the two A‟s and two R‟s do not occur together.
AA RR NN EE G M T
I'm going to think of this as AR NN EE G M T
How many distict ways can this be arranged. I have treated AA and RR as just one item each.
So I think this is how many ways the letters can be arranged so that the 2Rs and 2As are together.
\(\frac{9!}{2!2!}=\frac{9!}{4}\) ways
Now how many ways can the original set be arranged.
\(\frac{11!}{2!2!2!2!}= \frac{11! }{16}\)
So the number of ways that the 2As and the 2Rs are not together is
\(\frac{11!}{16}-\frac{9!}{4}=2494800-90720 = 2402080\)
I think.