Professor Grinch's telephone number is 6328363. Mickey remembers the collection of digits but not their order, expect that he knows the first 6 is before the first 3. How many arrangements of these digits with this constraint are there?
I am not for sure how to answer this question.
There are $7$ digits, and of the $7!$ permutations, $2/5$ of the permutations have $6$ come before $3$. Furthermore, with two $6$'s and three $3$'s, the permutation overcounts the arrangements by $2!\cdot3!$ times.
So providing I didn't make a mistake, we have
$$\frac{2\cdot7!}{5\cdot2!\cdot3!}=168$$