arragements

atleast 1 character between Is=> Is should not be together.

we shall find out total arrangements and deduct cases where Is are together to get cases where 1s are not together.

total arrangements possible: 7!/(2!*2!)=1260 ways.
(there are 7 letters in all, 2 letters repeated twice.)

no of arrangements in which Is are together: consider the Is as a unit so there are in effect 6 letters with Ts repeated twice
so 6!/2!=360 ways.

no of arrangments where Is are not together: 1260-360= 900 ways.

C?

OA: C

Step 1: calculate all possible ways to arrange 7 letters including 2 same letters.
All possible ways to arrange 7 DIFFERENT letters is:
7!=5040

But, remember there are 2 same letters I. So 7! contains duplicate arrangements. For example,
GMATTII is the same as GMATTII (where two Is were switched position)

Therefore, all possible ways to arrange 7 letters including 2 same letters is:
7!/2=2520

Step 2: calculate all possible ways to arrange 6 different letters (when two Is are never seperated, they can be considered as ONE letter)
6!=720

Step 3: 2520-720=1800
This is all possible arrangements when two Is are separated.

Answer is D.