Question

In how many different ways can the letters of the word ARRANGE be arranged? If the two 'R's do not occur together, then how many arrangements can be made? if besides the two R's the two A's also do not occur together, then how many permutations will be obtained?

Answer 1

The word "ARRANGE" can be arranged in $\frac{7!}{2!\times 2!}=\frac{5040}{4}=1260$ ways.

For the two R's do occur together, let us make a group of R's taking from "ARRANGE" and permute them.

Then the number of ways $=\frac{6!}{2!}=360$.

The number ways to arrange "ARRANGE", where two "R's" will not occur together is $=1260-360=900$.

Also in the same way, the number of ways where two "A's" are together is $360$.

The number of ways where two "A's" and two "R's" are together is $5!=120$.

The number of ways where neither two "A's" nor two "R's" are together is $=1260-(360+360)+120=660$.