Question

The number of ways of arranging the letters of the word 'ARRANGE' so that neither 2A's nor 2R's occurs together are

• A. $900$
• B. $240$
• C. $660$
• D. $7!$

Answer 1

The correct option is C $660$

Total number of ways $=$ Particular $K$ things occurs together $+$ never occurs together

$\therefore$ Number of ways in which $2R$ never together $=\frac{5!}{2!}\times \frac{{}^6{P}_2}{2!}=900$

Number of ways in which $2A$ are together but not two, $=4!\times \frac{{}^5{P}_2}{2}=24\times 10=240$
Now there are in all $900$ arrangement.

Either the two $A$ are together or two $A$ are not together.
Since number of all such arrangements in which the two $A$ are together $=240$,

Therefore, Number of arrangement in which neither $2R^{\prime }s$ nor $2A^{\prime }s$ are together is $=900-240=660$