As each participant arrives, she places her gift in the collecting area and receives a numbered ticket. A matching numbered ticket goes into a box, bag, or basket.
1. (easiest) When it is time for the exchange, you draw the tickets one or two at a time and those numbers choose from the display. Or, pick up a gift, draw a ticket, and that is what she gets (fastest).
2. (wildest) When each ticket is drawn, person may choose either from the table or "steal" from someone earlier. That person then may take a new gift or "steal". Limit the number of times an item may change hands to two or three.
3. (more work, but makes distribution go faster) Place numbers on all gifts before exchange. No drawing needed, just have small groups of participants come up at a time to find their numbers.