String permutations: ways to reorder DONALDDUCK with restrictions:

Question

How many ways are there to reorder the string DONALDDUCK when the first letter can't be K and last letter has to be D?

My solution: $$\frac{8*8!}{3!}$$ (we have three repeating letters and the first slot can't be K because of the restriction)

Unfortunately, I noticed that my solution is apprently wrong and it should be $2!$ in the denominator as if we ignore one of the D letters that is already used in the last slot but that doesn't make sense to me.

What's the correct solution?

Answer 1

The last letter has to be $D$.

Hence the question is equivalent to

How many ways are there to reorder the string ONALDDUCK when the first letter can't be K

Consider the number of possible arrangement with no restricition. The answer would be $\frac{9!}{2}$

Now, consider the number of possible arrangement where the first letter must be $K$. we have $\frac{8!}{2}$

$$\frac{9!-8!}{2}=\frac{8 \cdot 8!}{2}$$