Find the number of different ways of arranging letters in the word ARR

1.	Find the number of different ways of arranging letters in the word ARRANGE. How many of these arrangements do not have the two R’s and two A’s together?
Answer» There are 7 letters in the word ARRANGE in which ‘A’ and ‘R’ repeat 2 times each. ∴ Number of ways to arrange the letters of word ARRANGE = \(\frac{7!}{2!2!}\) = 1260 Consider the words in which 2A are together and 2R are together. Let us consider 2A as one unit and 2R as one unit. These two units with remaining 3 letters can be arranged in = \(\frac{5!}{2!2!}\) = 30 ways. Number of arrangements in which neither 2A together nor 2R are together = 1260 – 30 = 1230

Find the number of different ways of arranging letters in the word ARRANGE. How many of these arrangements do not have the two R’s and two A’s together?

Answer»

There are 7 letters in the word ARRANGE in which ‘A’ and ‘R’ repeat 2 times each.

∴ Number of ways to arrange the letters of word ARRANGE = \(\frac{7!}{2!2!}\) = 1260

Consider the words in which 2A are together and 2R are together.

Let us consider 2A as one unit and 2R as one unit.

These two units with remaining 3 letters can be arranged in = \(\frac{5!}{2!2!}\) = 30 ways.

Number of arrangements in which neither 2A together nor 2R are together = 1260 – 30 = 1230