# $8$-digit numbers that can be formed by using all the digits $0,1,2,3,4,5,7,9$

If the total number of ways in which $$8$$-digit numbers can be formed by using all the digits $$0,1,2,3,4,5,7,9$$ such that no two even digits appear together is $$(5!)k$$, then $$k$$ is equal to?

There are three even digits ($$0,2,4$$) and five odd digits ($$1,3,5,7,9$$).

If odd digits are taking odd places then number of digits are $$5!\cdot4!$$.

If odd digits are taking even places then number of digits are $$5!\cdot3\cdot3\cdot2\cdot1$$.

So, the total number of digits are $$(5!)42$$.

But the answer is given as $$100$$.

Edit: Elaborating my reasoning for the above calculations:

For considering odd or even, I am starting from the left of the numbers. i.e. in the number $$abcd$$, $$a$$ and $$c$$ are at odd places, $$b$$ and $$d$$ are at even places.

If five odd digits ($$1,3,5,7,9$$) are taking four even places they can be arranged in $${5\choose4}\cdot4!$$ ways. So, the remaining one odd digit and the three even digits can be arranged in $$4!$$ ways at even places. So, here, total numbers are $$5!\cdot4!$$.

If the odd digits are being placed at even places, they can be arranged in $${5\choose4}\cdot4!$$ ways. So, the remaining one odd digit and the three even digits would occupy odd places but $$0$$ can't be placed at the first place. So, that place can be filled only in $$3$$ ways. So, number of ways for filling the third place are $$3$$, for fifth place $$2$$ and for the seventh place $$1$$. So, here the total number of digits are $$5!\cdot3\cdot3\cdot2\cdot1$$.

So, total ways $$=5!\cdot42$$.

There are $$5$$ odd digits, but only $$3$$ even digits. Therefore they can't just alternate between odd and even.
You can position the odd digits in so many ways. These odd digits $$\bullet$$ produce $$6$$ spaces $$-$$ in between them and at the ends. Each of these spaces can receive at most one even digit. But you are forbidden to place the $$0$$ in front. Therefore place first legally the $$0$$, then the $$2$$ then the $$4$$. $$-\bullet-\bullet-\bullet-\bullet-\bullet-$$