Number of ways to divide $20$ distinct objects into five groups of size $6,6,6,1,1$

Question

Number of ways to divide $20$ distinct objects into five groups of size $6,6,6,1,1$

For this logically the answer should be

$$\frac{20!}{(6!)^3} \times \frac{5!}{3!2!}$$ Since the quintuple $6,6,6,1,1$ can be arranged in $\frac{5!}{3!2!}$ ways.

But my book answer is $$\frac{20!}{(6!)^3 \times 5!}$$

Why is it So?

Answer 1

If group size and membership are the only things that matter, then I think the answer is $\frac{20!}{(6!)^3 \cdot 3! 2!}$. If you want to arrange the 5 groups in a line, then you need to multiply by $5!$, which is your answer. I'm not sure how to interpret the book's answer.

Answer 2

I would say neither your answer nor the book's answer is correct.

Let's use the typical method of lining up all objects ($20!$ ways to do that), and then splitting them into the 5 groups by puting divisors between 6 and 7, 12 and 13, 18 and 19, and between 19 and 20. So now consider the ways in which we overcount:

First, we can swap any of the members in any group, so we need to divide by $(6!)^3$

Second, we can swap any groups of the same size, so we need to divide by $3!2!$

So, the final answer is:

$$\frac{20!}{(6!)^33!2!}$$