How many ways "ARRANGE" can be arranged?

Question

In how many ways can the letter of the word "ARRANGE" be arranged in which two Rs and two As come together?

Answer 1

We have to arrange all the letters of word ARRANGE such that AA and RR come together.

So, Put both A's and R's together,
i.e. consider AA and RR as single entities.

n items can be arranged in n! ways:
Proof:
$1^{st}$ item has n options, $2^{nd}$ item has (n-1) options,....$n^{th}$ item has 1 option.
So, total ways are:$$n\cdot(n-1)\cdot(n-2)\cdot\cdot\cdot1$$ i.e. n! ways.

You have AA,RR,N,G,E i.e. 5 entities to be arranged.
You can arrange 5 items amongst themselves in 5! ways.

So, final answer is 5!=120 ways.