Is a set of objects starting with zero velocity gravitationally bounded? Given a random set of objects in 3D space each with zero velocity and each with mass $M$. Now let them move under gravity. They will collapse in on each other. But will they then fly appart? Or is there enough mass in the combined system to keep any element from flying off to infinity and reaching escape velocity from the rest of the objects? Is there an intuitive proof?
Edit:Assume they are point-like.
 A: The answer to your either/or question is yes and yes. They will in general orbit around their centre of mass in charming and elaborate ways, but it is possible for encounters between the masses to lead to one exceeding the escape velocity of the cluster, and escaping. 
This actually does happen in globular clusters of stars - over time a star will “evaporate” from the cluster. 
Moreover, in a really pathological arrangement of masses (so very much non-random) it is possible for one of them to be projected arbitrarily far away in a finite time. I think the minimum number of masses for this is 5, but it may be less. Basically the gravitational potential energy of the “mass at infinity” comes from the gravitational potential energy lost by two of the other masses spiralling ever closer together. 
But for a random arrangement, the globular star cluster is the model. 
A: You don't say how many objects in your random set, so let's say it is five. You don't specify how far apart they are, so let's say they are 30 metres apart They each have mass M, and M happens to be a million tons. We now let them move under the influence of gravity, and as you say, they will collapse on each other. They have a friable, sandy or gravelly structure with a small admixture of ice. so when they collide they merge at their extremities. We now have something similar to a small comet. No, they won't bounce off each other, separate, and go flying off into space. The only proof you need to understand this is common sense, which is partly but not entirely intuitive.
A: In general the answer is yes. But depending upon initial relative positions and masses, they will start accelerating towards the common center of mass.
In this process, some of them will collide, and some of them will miss others in the process of gravitating towards center of mass.
In case of collisions, the results will be influenced by respective bouncing properties, masses and relative velocities. So let us exclude collisions from the scope because it would require more information than is given. 
In cases where they miss others (i.e do not collide), they will gravitate towards orbits. In this process some can escape the system - think of slingshot. It would be a conceptually simple, but mathematically complex process depending upon initial conditions.
