1. We need to count the species having trigonal bipyramidal molecular shape.

   Given species: $\mathrm{PF_5},\ \mathrm{BrF_5},\ \mathrm{PCl_5},\ \left[\mathrm{PtCl_4}\right]^{2-},\ \mathrm{BF_3},\ \mathrm{Fe(CO)_5}$

2. Check each species one by one.

(i) $\mathrm{PF_5}$

• Central atom: P
• Valence shell arrangement: 5 bond pairs, 0 lone pairs
• VSEPR type: $AX_5$
• Shape: trigonal bipyramidal

So, $\mathrm{PF_5}$ is counted.

(ii) $\mathrm{BrF_5}$

• Central atom: Br
• Total electron pairs around Br: 5 bond pairs + 1 lone pair
• VSEPR type: $AX_5E$
• Electron pair geometry: octahedral
• Molecular shape: square pyramidal

So, $\mathrm{BrF_5}$ is not counted.

(iii) $\mathrm{PCl_5}$

• Central atom: P
• 5 bond pairs, 0 lone pairs
• VSEPR type: $AX_5$
• Shape: trigonal bipyramidal

So, $\mathrm{PCl_5}$ is counted.

(iv) $\left[\mathrm{PtCl_4}\right]^{2-}$

• Pt is a $d^8$ metal ion
• This complex is well known to be square planar

So, $\left[\mathrm{PtCl_4}\right]^{2-}$ is not counted.

(v) $\mathrm{BF_3}$

• Central atom: B
• 3 bond pairs, 0 lone pairs
• VSEPR type: $AX_3$
• Shape: trigonal planar

So, $\mathrm{BF_3}$ is not counted.

(vi) $\mathrm{Fe(CO)_5}$

• Coordination number of Fe = 5
• The structure of iron pentacarbonyl is trigonal bipyramidal

So, $\mathrm{Fe(CO)_5}$ is counted.

3. Total trigonal bipyramidal species: $\mathrm{PF_5},\ \mathrm{PCl_5},\ \mathrm{Fe(CO)_5}$

Hence, number of species having trigonal bipyramidal shape is $3$

4. Comparison with stored correct answer:

• Derived answer = $3$
• Stored correct answer = $3$
• They match.