1. Judas went to the Chief priests to betray Jesus for money (Matt. 26:14; Mark 14:10; Luke 23:2-6).
2. The disciples all ate at least the initial part of the Passover Supper (Mark 14:17; Luke 22:14).
3. After the intial part of the Passover Supper was ended, the devil puts it in Judas' heart to betray Jesus; Jesus washes all their feet (John 13:2-11).
4. They were told that they were not all clean, a clear reference to Judas. (John 13:11). 5. All, or a few of them, sat back down at the table to dip sops. (John 13:12, 28).
6. The disciples are then told that one of them would betray Jesus. (John 13:21).
7. They ask, including Judas, "Is it I?" and who it was. (Matt. 26:25; John 13:22-25).
8. They are told that the one that dips his 'sop' with Jesus would be His betrayer. (John 13:26; Mark 14:18-20).
9. Judas is told that he is the one and given the 'sop.' (John 13:27, 30).
10. After being given the sop, Judas IMMEDIATELY goes out from them and goes to the chief priests. (John 13:30).
11. Jesus institutes the Lord's Supper/Christian Communion with the eleven. (Matt. 26:26; Mark 14:22-25; Luke 22:19-20). Judas is no longer present.
12. Jesus Expounds upon the meaning of His life and the Gospel (John 13:31 to John 18:1).
13. The eleven sing a Hymn (Mark 14:46).
14. The eleven go out to the Mount of Olives and the Garden of Gethsemane (John 18:1; Luke 22:39).
15. Judas takes Jesus' enemies to Him, betraying him with a kiss (Matt. 26:47; Mark 14:43; Luke 22:47, 48; Acts 1:16; John 18:2-5).
16. Judas repents and hangs himself (Matt. 27:3; Acts 1:25).
Looks like a contradiction. Only if Scripture isn't looked at in its totality. Luke typically didn't record events chronologically.
Without high-jacking the thread, let me say that this explanation is certainly plausible. Others have pointed out that the other Gospel accounts can be viewed in a way to reconcile them to Luke's account. This is an apparent contradiction whose resolution isn't universally accepted. I think that there is enough uncertainty that dogmatism about Judas' presence or absence should be avoided.
It has no effect on my view of the Lord's Supper. There will be people who will improperly take the Lord's Supper and who will be punished for doing so as Paul instructed in 1 Corinthians 11. It is possible that Judas was the first.