Well I looked at it like this.
1K and 5V gives you 5mA current down the collector.
10K and 5V gives you 0.5mA through the base.
So the minimum gain the transistor needs is 5 / 0.5 or 10
Any transistor worth it's salt can manage a gain of 10, most small signal transistors have a gain of 100 or so. Therefore it will be fully saturated.
You do not need the currents that DVDdoug is suggesting, in fact that will only make the transistor slower to switch.