How many hybrid orbitals are present in the molecule PCl5?

There are either zero or 15 hybrid orbitals present in the PCl₅ molecule.

A better question would be, "How many hybrid orbitals does a P atom use when it forms a molecule of PCl₅?

theory predicts that PCl₅ should have a trigonal bipyramidal structure. This corresponds to an sp³d hybridization.

The P atom has five hybridized orbitals.

Once the PCl₅ molecule has formed, the P atom no longer has these hybrid atomic orbitals. They have overlapped with orbitals of Cl atoms to form σ molecular orbitals. You usually see the picture below, in which an sp³d orbital overlaps with an unhybridized p orbital of Cl to form a σ molecular orbital.

In this model, the PCl₅ molecule has no hybridized orbitals.

I suggest that this model is incorrect. If you apply VSEPR theory to one atom in a molecule, you must apply it to the other atoms as well.

Each Cl atom has four electron domains surrounding it. VSEPR theory says they will arrange themselves in a tetrahedral geometry. This corresponds to an sp³ hybridization.

Each Cl uses an sp³ orbital to bond to the P. The remaining three orbitals contain the lone pairs of Cl.

The five Cl atoms then have fifteen hybridized orbitals.