Probability each of six different numbers will appear at least once?

Question

If seven balanced dice are rolled, what is the probability that each of the six different numbers will appear at least once?

My solution is $\frac{6 \cdot 7!}{6^7}$.

Is this correct? How would I implement a solution using multinomial coefficients?

Answer 1

There are $6^7$ sequences of $7$ tosses, all equally likely.

We now count the sequences in which all the numbers appear. The number that appears twice can be chosen in $6$ ways. For every such choice, where it appears can be chosen in $\binom{7}{2}$ ways. And the rest of the numbers can be put in the empty slots in $5!$ ways. So our probability is $$\frac{(6)\binom{7}{2}(5!)}{6^7}.$$