Case 1: No repeated letters from these 5: {I,N,K,L,G}

Choose the 1st letter 5 ways.
Choose the 2nd letter 4 ways.
Choose the 3rd letter 3 ways.
Choose the 4th letter 2 ways.

That's 5P4 = 5*4*3*2 = 120 ways for case 1.

Case 2: There are 2 I's but not 2 N's
Choose the 2 positions for the I's 4C2 = 6 ways
Choose the letter for the leftmost unchosen position 4 ways, from {N,K,L,G}
Choose the letter for the rightmost unchosen letter 3 ways.

That's 6*4*3 = 72 ways for case 2.

Case 3: There are 2 N's but not 2 I's
Same as Case 2.

That's also 72 ways for case 3.

Case 4.  2 N's and 2 I's
Choose the positions for the N's 4C2 = 6 ways.
Put I's in the remaining 2 places only 1 way.

That's 6*1 = 6 ways for case 4

Grand total: 120+72+72+6 = 270

Answer: Choice (A)270

Edwin