in how many ways can the letters of mahabharat be arranged so that, b and t are never together and m,a,t,h occurs at first four places
There are 4! ways to arrange m,a,t,h in the 1st 4 places.
Of these, 3! have t as the last letter.
The remaining letters are aaabhr, which can be arranged in 6!/3! ways.
So, we have (4!-3!)(6!/3!) ways with t not in the 4th place.
Add to that (3!)(6!/3! - 5!/3!) ways of having t in 4th place and b not in 5th place.
Looks like 2760 ways to me.
If that's incorrect, I'm sure you can find the flaw.