In how many ways can the letters of the word ‘ARRANGE’ be arranged

1.	In how many ways can the letters of the word ‘ARRANGE’ be arranged such that the two r’s do not occur together?
Answer» There are two a’s, two r’s in the word ‘arrange’, therefore the number of arrangements = \(\frac{7!}{2!2!}=1260.\) ...(1) The number of arrangements in which the two r’s occur together = \(\frac{6!}{2!}\) = 360 ...(2) ∴ The number of arrangements in which two r’s do not occur together = (1) – (2) = 1260 – 360 = 900.

In how many ways can the letters of the word ‘ARRANGE’ be arranged such that the two r’s do not occur together?

Answer»

There are two a’s, two r’s in the word ‘arrange’, therefore the number of arrangements = \(\frac{7!}{2!2!}=1260.\) ...(1)

The number of arrangements in which the two r’s occur together = \(\frac{6!}{2!}\) = 360 ...(2)

∴ The number of arrangements in which two r’s do not occur together = (1) – (2) = 1260 – 360 = 900.