Question

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

Answer 1

There are 7 letters in hte word 'ARRANGE' out of which 2 are A's 2 are R's and the rest are all distinct.

So, total number of words = $\frac{7!}{2!2!}$

= $\frac{7\times 6\times 5\times 4\times 3\times 2!}{2\times 2!}$

=$7\times 6\times 5\times 2\times 3$

= 1260
Considerting all R's together and treating them as one letter we have 6 letters out of which A repeats 2 times and other are distinct.

These 6 letters can be arranged in $\frac{6!}{2!}$ ways.

So, the number of words in which all R's come together =$\frac{6!}{2!}$

= $\frac{6\times 5\times 4\times 3\times 2}{2!}$

= 360.

Hence, the number of words in which all R's do not came together

= Total number of words - Numbers of words in which all R's come together

= 1260-360

= 900