1. Interpret the given name

The compound is named:

$\text{dibromidobis(ethylenediamine)chromium(III) bromide}$

Let us decode this systematically.

• dibromido $\Rightarrow$ there are 2 coordinated bromide ligands inside the coordination sphere.
• bis(ethylenediamine) $\Rightarrow$ there are 2 ethylenediamine ligands inside the coordination sphere.
• chromium(III) $\Rightarrow$ oxidation state of chromium is $+3$.
• bromide written at the end $\Rightarrow$ there is one bromide ion outside the coordination sphere as counter ion.

So the complex should be of the form:

$[Cr(en)_2Br_2]Br$

2. Check charge on the complex ion

Now verify the charge.

• Oxidation state of $Cr = +3$
• Each $en$ ligand is neutral.
• Each coordinated $Br^-$ contributes $-1$

Hence charge on the coordination entity:

$+3 + 2(0) + 2(-1) = +1$

So the complex ion is:

$[Cr(en)_2Br_2]^+$

To balance this, one bromide ion outside is required:

$[Cr(en)_2Br_2]Br$

This matches the name exactly.

3. Evaluate the options

Option A

$[Cr(en)_3]Br_3$

This has three ethylenediamine ligands and no coordinated bromido ligands. So its name would not be dibromidobis(ethylenediamine)chromium(III) bromide.

❌ Incorrect

Option B

$[Cr(en)_2Br_2]Br$

This has:

• 2 bromido ligands inside
• 2 ethylenediamine ligands inside
• 1 bromide counter ion outside

Exactly matches the given name.

✅ Correct

Option C

$[Cr(en)Br_4]^-$

This has:

• 1 ethylenediamine ligand
• 4 bromido ligands
• an anionic complex

It does not match the given name.

❌ Incorrect

Option D

$[Cr(en)Br_2]Br$

This has only:

• 1 ethylenediamine ligand
• 2 bromido ligands

The name would involve mono(ethylenediamine), not bis(ethylenediamine).

❌ Incorrect

4. Final answer

The correct compound is:

$[Cr(en)_2Br_2]Br$

So the correct option is B.