Question

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 1

In the word ARRANGE the number of letters is n = 7 of which A repeats twice, i.e., p = 2, R repeats twice, i.e., q = 2 and the rest are distinct.

∴ the number of ways in which the letters of the word 'ARRANGE' are arranged

= `(n!)/(p! q!)`

= `(7!)/(2! 2!)`

= `5040/4`

= 1260

Neither two R's nor two A's are together:

Let two R's and two A's occur together. Considering two R's as one unit and two A's as another unit we have 3 + 2 = 5 letters.

∴ number of arrangements of letters in which two R's and two A's are together = `(5!)/(2! 2!)`

= `(5 xx 4 xx 3 xx 2 xx 1)/(2 xx 1 xx 2 xx 1)`

= 30

∴ number of arrangements in which neither two R's nor two A's are together

= 1260 – 30

= 1230