No, the house does not go through probate. The inter vivos (or "living") trust document controls and once your mother passed away, the trust property automatically went to the named beneficiaries of the trust. Of course, this answer is based on the facts you have given. If the trust document says the will controls or any language that would indicate otherwise, then it is possible the property would have ot go through probate. Normally, however, the purpose of putting any property in an inter vivos trust is to avoid having it go through probate. You may want to have a lawyer read over the trust document if you are uncertain.
The beauty of real estate held in the name of a living trust is that it avoids probate. Since the four children were named as the beneficiaries the property can be transferred directly to them.