What is the permutit method, how can it be used for softening hard water?

Hint: Permuted method is used for filtration of hard water. Filtration removes the impurities. Permutit is an artificial zeolite. Permutit has a porous structure. The pores can trap ions of specific size.

Complete step by step solution:
Based on the concentration of ions particularly calcium and magnesium, the water is divided into two parts: Hard water and soft water.
Hard water: the water contains a high amount of dissolved salts.
Soft water: the water contains a low amount of dissolved salts.
Hard water has a multivalent cation. The commonly present cations are calcium ${\text{C}}{{\text{a}}^{{\text{2 + }}}}$ and magnesium ${\text{M}}{{\text{g}}^{{\text{2 + }}}}$. The ions enter into the water for leaching. The ions are mixed into the water in the form of their minerals.
The permutit method is a simple filtration method. In this method zeolite that is also known as natural sand is used to filter the hard water.
Hard water is passed through sodium permutit. Permutit is a hydrated sodium alumina-orthosilicate. The formula of permutit is ${\text{N}}{{\text{a}}_{\text{2}}}{\text{A}}{{\text{l}}_{\text{2}}}{\text{S}}{{\text{i}}_{\text{2}}}{{\text{O}}_{\text{8}}}{\text{.x}}{{\text{H}}_{\text{2}}}{\text{O}}$. It has a porous structure.
In this method, permutit is taken into a cylinder and hard water is passed over it. Ions get exchanged. The calcium and magnesium get trapped in zeolite and sodium ion passes from the zeolite to water thus, hard water becomes soft as the calcium and magnesium ion removes from water and sodium permutit converts into calcium and magnesium permutit.

Note: Zeolites are a shape-selective catalyst. Zeolites have cavities of specific size, so they can trap ions according to the size of the cavity. During the permuted method, the zeolite permutit gets deactivated so it regenerates by passing a concentrated brine solution over it. So, calcium and magnesium ions get converted into chloride, by washing these chlorides removes and regenerates the permutit.