[talanum1 (OP)]

All of Trappist-1's seven planets orbits in a 3:2 ratio. How did that happen by chance?

[Origin]

All of Trappist-1's seven planets orbits in a 3:2 ratio.

They don't.

[evan_au]

There are orbital patterns of two objects which are stable.
- There are some specific orbital patterns of 3 objects which are stable (eg Lagrange Points L4 & L5)
- But in general, any gravitational system or 3 or more objects has no closed-form mathematical solution, and must be modelled numerically.
- All the orbiting planets continually exchange momentum between them. At some points in time, this momentum becomes large enough to force one or more objects out of their orbits - perhaps changing places, or one expelled from orbit while the other orbit becomes very elliptical.
- Some orbits are particularly unstable - such as a 2:1 ratio, as the planets attract each other when they are the same point on every orbit.
- The orbits in our Solar System are semi-stable, which have been modeled out to 100 million years (ignoring outside influences)

Even if Trappist started with 3:2 orbital periods, they would not stay that way forever.

See: https://en.wikipedia.org/wiki/TRAPPIST-1