find the no. of ways in which letters of the word ARRANGEMENT can be arranged so that 2A's and 2R's do not occur together?

Answer 1

number of arrangements without any restriction
= 11!/(2!2!2!2!) ----- 2 R's, 2 A's, 2 E's and 2N's
= 2494800

Assume the 2R's and the 2A's as unique elements
so we have 9 elements to arrange, but we still have the 2 E's and the 2 N's
number of ways to arrange = 9!/(2!2!) = 90720

the number of ways for the A's and the R's to be apart
= 2494800-90720 = 2404080