the solution of a problem is this: Total number of words = 5! = 120! If all the vowels come together, then we have: (O.E.A.), M,G These can be arranged in 3! ways. But (O,E.A.) can be arranged themselves in 3! ways. => Number of ways, when vowels come-together = 3! x 3! = 36 ways => Number of ways, when vowels being never-together = 120-36 = 84 ways. but why not, like this:- O X E X A over here the position of X can be filled by 2! ways and OEA in 3! ways,what more selections am i missing? becoz total ways according to my analysis is 12 but according to this process is 84.

the solution of a problem is this:

Total number of words   =   5!   =    120!

 If all the vowels come together, then we have: (O.E.A.), M,G

 These can be arranged in    3!    ways.

 But (O,E.A.) can be arranged themselves in   3! ways.

 => Number of ways, when vowels come-together  =    3!  x    3!  

= 36 ways

=> Number of ways, when vowels being never-together

= 120-36        =  84 ways.

but why not,

like this:-

O  X  E X   A 

over here the position of X can be filled by 2! ways and OEA in 3! ways,what more selections am i missing?

becoz total ways according to my analysis is 12 but according to this process is 84.