Question

In how many ways can the letter of the word "ARRANGE" be arranged so that
(i) the two R's are never together?
(ii) the two A's are together but not the two R's?
(iii) neither the two A's nor the two R's are together?

Answer 1

There are total 7 letters with 2 'A' and 2 'R'.

Total number of arrangements$=\frac{7!}{2!2!}=1260$

$\left(i\right)$ When two 'R' are together, taking both $R$ as one entity, number of different elements are $7-2+1=6$ and as two 'R' are same there will be no self arrangement within the entity.

Number of arrangement with 2 'R' together is $\frac{6!}{2!}=360$

Hence number of arrangement with no two 'R' together$=1260-360=900$

$\left(ii\right)$ The number of arrangement with two 'A' together is $\frac{6!}{2!}=360$

The number of arrangement of both two 'A' and two 'R' together is -

Take group of two 'A' as one and two 'R' as another entity .

Number of elements left is now $7-4+2=5$ and the 'R' and 'A' being same are not arranged within entity.

Number of arrangement with two 'A' together and two 'R" together is $5!=120$

Hence, number of arrangement with two 'A' together but two 'R' not together

= number of arrangement with 'A' together-number of arrangement with 'A' as well as 'R' together

$=360-120=240$

$\left(iii\right)$ Number of arrangement with neither 'A' nor 'R' together

= total number of arrangement$-$number of arrangement with only 'A' together not 'R'$-$number of arrangement with only 'R' together not 'A'$-$ number of arrangement with both 'A' and 'R' together grouped

$=1260-240-240-120$

$=660$