What is the order of the strengths of the three Bronsted acids.?

A stronger Bronsted acid can react with a base to form a weaker Bronsted acid.
However, a weaker Bronsted acid cannot react with a base to form a stronger Bronsted acid.

Reaction 1: CH₃NH₃⁺ + OH⁻ → H₂O + CH₃NH₂ goes
The Bronsted acid CH₃NH₃⁺ can neutralize OH⁻ ion to form the Bronsted acid H₂O.
Hence, CH₃NH₃⁺ is a stronger Bronsted acid than H₂O.

Reaction 2: (CH₃)₂NH₂⁺ + CH₃NH₂ → CH₃NH₃⁺ + (CH₃)₂NH does not go
The Bronsted acid (CH₃)₂NH₂⁺ cannot neutralize CH₃NH₂ to form the Bronsted acid CH₃NH₃⁺.
Hence, CH₃NH₃⁺ is a stronger Bronsted acid than (CH₃)₂NH₂⁺.

Reaction 1: (CH₃)₂NH₂⁺ + OH⁻ → H₂O + (CH₃)₂NH goes
The Bronsted acid (CH₃)₂NH₂⁺ can neutralize OH⁻ ion to form the Bronsted acid H₂O.
Hence, (CH₃)₂NH₂⁺ is a stronger Bronsted acid than H₂O.

Hence, acid strength: CH₃NH₃⁺ > (CH₃)₂NH₂⁺ > H₂O
OR: acidic strength: H₂O < (CH₃)₂NH₂⁺ < CH₃NH₃⁺