How many ways can you arrange the word KOLKATAA, such that no two K or A come together

3! X ⁴c₃ X 1! X ⁷c₂ X 1! 

This can be obtained by first arranging O L and T in 3! Ways and then choosing any 3 places for A between them out of 4 in ⁴c₃ ways and permuating them amongst themselves . And finally choosing any 2 places for K out of 7 in between of A  O L and T  and again permuating them against themselves.