one the only way (changed number in bold):
71549 -> 71569 -> 71069 -> 11069 -> 10069 -> 10067
First number change:
The only possible (read: still prime) switches are 11549, 70549, and 71569
Second number change:
11549 does not have any possible moves, so that number will not work. The same goes for 70549. Therefore, the only number from the first move that still applies is 71569. The only number that is still prime is 71069.
Third number change:
Taking 71069, the only possible move is 11069.
Fourth number change:
Taking 11069, only 10069 works.
Last number change:
We move on the 10067, the final prime. There are no other ways to get to this number from 71549, changing only one number at a time AND keeping every intermediate value prime.