In how many ways the letters in the word MANAGEMENT can be arranged such that A’s does not come together.

• Part 1:
  The word MANAGEMENT can be arranged in 10! ways if we don't take any constraints.
  
  Part 2:
  Now consider the case of arranging the letters of the word MANAGEMENT such that, in all its arrangements, both A's will appear together. Like: MNGEMENT(AA), M(AA)NGEMENT, (AA)MNGEMENT, etc. Assume both A's to be one single character $. So it would be interpreted something like MNGEMENT$, M$NGEMENT, $MNGEMENT, etc. These set of 9 characters, can be arranged in 9! ways.
  
  Part 3:
  Now subtracting both the values of "All possible arrangements" and "Arrangements of both A's together" would give
• 13 years agoHelpfull: Yes(9) No(15)
• there are total 10 letters..
  no. of M = 2
  no. of A = 2
  no. of N = 2
  no. of E = 2
  and others appear only one...
  no. of ways to arrange these 10 letters = 10! / [ 2! * 2! * 2! * 2!]
  = 226800
  now, assume that two A i.e AA as single letter, total 9 letters
  no. of ways to arrange these 9 letters where two 'A' always together
  = 9! / [ 2! * 2! * 2! ]
  = 45360
  
  no. of ways of arrangement where two 'A' do not come together = 226800-45360
  = 181440
• 10 years agoHelpfull: Yes(9) No(0)
• since, there are 8 letters except A, hence we can place A in 9 different position
  so, the no. of ways = 9p2 * 8p8 / 2p2 * 2p2 * 2p2 * 2p2 = 181440
• 9 years agoHelpfull: Yes(0) No(0)