In how many ways can the letters of the word ‘ARRANGE’ be arranged so that the two R’s are never together?

There are

   

letters in the word

   

out of which

   

and the rest all are distinct.

So by using the formula,

   

total number of arrangements

   

   

Or,

   

   

Let us consider all R’s together as one letter, there are

   

letters remaining. Out of which

   

times A repeats and others are distinct.

So these

   

letters can be arranged in

   

Ways.

The number of words in which all

   

come together

   

   

Or,

   

   

So, now the number of words in which all L’s do not come together = total number of arrangements – The number of words in which all L’s come together

   

   

Hence, the total number of arrangements of word

   

 in such a way that not all R’s come together is