In how many ways can the letters of the word 'ARRANGE' be arranged? How many of these arrangements are in which (1) two R's come togethe

Sure, I can help with that.

First, let's find out how many ways the letters of the word 'ARRANGE' can be arranged.

The word 'ARRANGE' has 7 letters, but the letters 'A', 'R' and 'N' are repeated.

The formula for permutations of a multiset (where some items are repeated) is n! / (r1! * r2! * ... * rk!), where n is the total number of items, and r1, r2, ..., rk are the numbers of each type of item.

In this case, n = 7 (the total number of letters), r1 = 2 (the number of 'A's), r2 = 2 (the number of 'R's), and r3 = 2 (the number of 'N's).

So the total number of arrangements is 7! / (2! * 2! * 2!) = 2,520.

Now, let's find out how many of these arrangements have the two 'R's together.

If we consider the two 'R's as a single item, then we have 6 items to arrange ('A', 'A', 'N', 'N', 'G', 'E', and 'RR').

So the total number of arrangements is 6! / (2! * 2!) = 360.

But, the 'RR' can be arranged in 2! = 2 ways.

So the total number of arrangements where the two 'R's are together is 360 * 2 = 720.