When it is necessary to count by dividing possibilities into "cases", you generally want to identify the aspect of the solutions that is most constraining.
Here we have four $B$'s and five positions into which those $B$'s can be placed. That can be done in 5 ways, so it's a good starting point if we have to "divide into cases":
Original word: BEBBEDREB
Possible case: -B--BBB?-
Possible case: -B--BB?B-
Possible case: -B--B?BB-
Possible case: -B--?BBB-
Possible case: -?--BBBB-
We then have to pursue each case further, but it may be that some cases are alike and simplify our counting on that basis. Here we have assigned locations to the $B$'s in each of the 5 cases, and what remains is to assign the non-$B$ characters, three $E$'s and one $D$ and one $R$.
What goes in the
? position indicated above depends on what was there in the original word, as the new symbol must differ from the original one. In the 3 cases where the original symbol was $E$, the two choices are $D$ and $R$ for the new word in that position. Further, once that choice is made, all the remaining arrangements are legitimate, in that the last four positions were originally $B$'s and will now be assigned non-$B$ symbols.
So for these 3 cases, we choose 2 possible symbols ($D$ or $R$) for the question mark position, and then we have any possible arrangement of the three $E$'s and one non-$E$ symbol (whichever $D$ or $R$ is left from that last choice). Now the three $E$'s go into the four locations in 4 ways (basically just choose the location of the non-$E$).
Altogether these cases have 3*2*4 solutions.
That leaves the other two cases of the 5 outlined above, where the
? occurs under the $D$ or under the $R$. Again we have two choices for the symbol to be placed there, either an $E$ or the "opposite symbol" ($D$ in place of $R$ or vice versa). Any arrangement of the remaining four symbols is again legitimate (since they will go in slots previously occupied by $B$'s), but there is a slight twist.
If we choose $E$ to go in the
? slot, we are left with two $E$'s, one $D$ and one $R$, and there would be 12 ways to arrange those four symbols (think about picking a location for $D$ and then one for $R$). But if we choose $D$ or $R$ as "opposite" to go in the
? slot, we are left with three $E$'s and one non-$E$ symbol, and so as before 4 ways to arrange them.
Altogether for these last 2 cases we thus have 2*(12+4) solutions.
Combining the results for all 5 cases, there are 3*2*4 + 2*(12+4) = 56 solutions.