A different approach in distributing $8$ distinct balls into $6$ distinct boxes such that each box has at least $1$ ball

Question

Find the number of ways in distributing $8$ distinct balls into $6$ distinct boxes such that there is at least $1$ ball in each box.

We are well acquainted with the traditional Inclusion-Exclusion principle in solving this and it will come up as $191520$ ways.

I tried in this way:

When we distribute $8$ balls into $6$ boxes we have two possible cases

Case $1.$ There are exactly $2$ boxes with $2$ balls each

Case $2.$ There is exactly $1$ Box with $3$ balls in it.

For Case $1$, the distribution is done by choosing $2$ boxes from $6$ boxes and $2$ balls from $8$ balls and those two balls can be arranged in $2!$ ways. But I require another two balls from remaining $6$ balls to be placed in each of these two boxes which can be done in $\binom{6}{2} \times 2!$ ways. Finally remaining $4$ balls can be distributed in remaining $4$ boxes in $4!$ ways. So total number of ways for Case $1.$ is

$$\binom{6}{2} \times \binom{8}{2} \times 2! \times \binom{6}{2} \times 2! \times 4!=604800$$

For Case $2$, the number of ways is

$$\binom{6}{1} \times \binom{8}{3} \times 5!=40320$$

I am pretty sure the value obtained for Case $2.$ is correct. but I could not find where I went wrong in Case $1.$

Answer 1

On the case 1 you must choose first $2$ boxes over $6$ i.e. $C(6,2)$ and after for the first box you choose $(8\cdot 7)/2!$ and after $(6\cdot 5)/2!$ for the second box, and in third place you count the permutations of the others boxs that is $4!$, so the total is $\binom{6}{2}^2\cdot \binom{8}{2}\cdot4!=15^2\cdot 28\cdot 24=151200$.

And $151200+40320=191520$ as stated.

Answer 2

Case 1 is wrong: you pick 2 boxes from 6, in $\binom{6}{2}$ ways. These boxes are different, say boxes $i < j$. First fill box $i$ with 2 balls out of 8, so $\binom{8}{2}$ ways. Then fill box $j$ with 2 out of 6, so $\binom{6}{2}$ ways. Then we are left $4!$ ways for the remaining 4 balls in the remaining 4 boxes.

It only matters that balls say 7 and 5 are in box $i$, there is no order within that box! So we get a number 4 times as small, and then it checks out with your other answer.