I thought the answer was 5!x4! - 4!x3!
Let's say we group (NCOUT) in this specific order and then join with other letters N, I, and G to end up with (NCOUT)-N-I-G i.e. "NCOUTNIG"; There is another way among 5!x4!: we can group (COUTN)in this specific order which still contains letters from "COUNT". then we join (COUTN) with other three letters N, I, and G to end up with N-(COUTN)-I-G, i.e. "NCOUTNIG". This is one example of overcounting. (COUT) has 4! ways to rearrange in this order. After that "NCOUTN", I and G can be rearranged in 3! ways. so the overcounting happened 4!x3! times.
