The correct order of increasing the number of valence shells among Fe, Hs, Ru and Os is:

The correct order of increasing number of valence shells would be: Fe < Ru < Os < Hs.

To determine the correct order of increasing the number of valence shells among Fe, Hs, Ru, and Os, we need to look at the electron configurations of these elements.

Fe (Iron): [Ar] 3d⁶ 4s²
Hs (Hassium): [Rn] 5f¹⁴ 6d⁶ 7s²
Ru (Ruthenium): [Kr] 4d⁷ 5s¹
Os (Osmium): [Xe] 4f¹⁴ 5d6 6s²

The number of valence shells is determined by the highest principal quantum number (n) in the electron configuration. In the case of transition metals, the outermost s and d orbitals are considered to be in the valence shell.

So, the correct order of increasing the number of valence shells is:

1. Fe (Iron) - 4th period (n=4) - 4 valence shells
2. Ru (Ruthenium) - 5th period (n=5) - 5 valence shells
3. Os (Osmium) - 6th period (n=6) - 6 valence shells
4. Hs (Hassium) - 7th period (n=7) - 7 valence shells

Therefore, the correct order is: Fe < Ru < Os < Hs.