Number of words formed using all the letters of the word 'EXAMINATION' if alike letters are never adjacent.

Explanation

We have word EXAMINATION
Letters are E,X,(A A),M,(II),(N N),T,O.
Let P=set of words in which A, A are together
Q=set of words in which E, E, are together
R=set of words in which N, N are together
`n(P cup Q cup R)`=number of words in which at least one of the pairs A A, II, N N together.
`=n(P)+n(Q)+n(R )-n(P cap Q)-n(Q cap R)-n(R cap P)+n(P cap Q cap R)`
`=(10!)/(2!2!)+(10!)/(2!2!)+(10!)/(2!2!)-(9!)/(2!)-(9!)/(2!)-(9!)/(2!)+8!`
`n(P' cap Q' cap R')=n(U)-n(P cup Q cup R)`
`=(11!)/(2!2!2!)-3xx(10!)/(2!2!)+3xx(9!)/(2!)-8!`