The dice is tossed $7$ times . It has 6 faces so possible sample space is $\large 6^{7}$ .
Favorable cases= no of choices after tossing 7 times all the faces appear at least once.
So The possible arrangement is $1,2,3,4,5,6,x$ . [ $x$ can be anything between $1-6$]
Favorable choices= $\large \binom{6}{1}*\frac{7!}{2!}$ [ choosing $x$ in $\large \binom{6}{1}$ ways then permute then in $\large \frac{7!}{2!}$]
Probability = $\large \frac{6*7!}{2! *6^{7}}$=$\large \frac{7!}{2! *6^{6}}$ -----[desired answer]