Answer
If God is the real Author of the Bible, then there should be a special reason why He selected four writers to describe the life of Jesus. We must not limit our study only to one writer. All the four writers together provide the whole picture.
So Matthew, Mark, Luke and John provide different bits of the jigsaw puzzle, all of which complete the full picture.
(Since Dan Fefferman was kind enough to mention my reference of a jigsaw puzzle, I thought a little addition on that wouldn’t be out of place).
The Word of the Lord a Jigsaw Puzzle
God Himself witnesses regarding His Word as a jigsaw puzzle:
“But the word of the LORD was unto them precept upon precept, precept upon precept; line upon line, line upon line; HERE a little, and THERE a little” (Isaiah 28:13).
Yes, God has scattered all the details of a single topic throughout the Scripture as “little(s)” here and there. Anyone who studies this must collect all the little and place them line upon line and the precept will become clear.
This is exactly why we get a new insight every time we read the Scripture.
An Example
A good example of this can be demonstrated using the inscription on the cross.
Matthew writes: “THIS IS JESUS, THE KING OF THE JEWS” (Mat 27:37).
Mark writes: “THE KING OF THE JEWS” (Mark 15:26).
Luke writes: “THIS IS THE KING OF THE JEWS” (Luke 22:38).
John writes: “JESUS THE NAZARENE, THE KING OF THE JEWS” (John 19:19).
Now, to some, these show contradictions in the Bible.
But if one applies the principle found in Isaiah 28:13, these are “here a little and there a little”. Place them line upon line, then the precept is very clear.
The exact word written on the cross is this:
“THIS IS JESUS THE NAZARENE, THE KING OF THE JEWS”.
There is no contradiction, at all!
The four Gospel writers were giving bits and pieces of the true inscription. These are not contradictions. We must take into account the details of all the four writers together.
Now The Sequence of the Tomb Incident
The women including Mary Magdalene reach the tomb and find the stone removed (John 20:1).
They enter the tomb (people are less afraid in groups) and find the body of Jesus missing (Luke 24:3).
They run to the place of disciples and inform them about the empty tomb (John 20:2). Peter and John run to the tomb, see the clothes inside the tomb and go back to their place (John 20:3-10). The other ladies accompany them to some distance to say good bye.
But Mary stands weeping at the entrance of the tomb alone. She sees two angels in the tomb. They talk to her (John 20:11-13). But she doesn’t go inside (alone she is afraid to enter).
Outside, Jesus meets her alone at first (John 20:14-17; Mark 16:9).
Mary finds the other women and tells them. They don’t believe her. All of them go to the tomb and enter it (Mark 16:5). (As a group, they find courage to enter the tomb). They see the angels again and get afraid (Luke 24:5). One of the angels tells them to inform the disciples especially Peter (Mark 16:7). They run trembling to the disciples’ place without talking to anyone on the way (Mark 16:8).
Trembling with fear and yet with great joy, they run to tell the disciples about the risen Jesus (John 20:18; Luke 24:9) but Jesus meets all of them together on their way (Mat 28:8-10).
The apostles don’t believe them (Mark 16:11; Luke 24:11).
But since Peter’s name was mentioned specially by the angels (Mark 16:7), he alone runs to the tomb a second time, sees the clothes again. But this time he goes back to “his own home, wondering” (Luke 24:12).
Conclusion
Mary with other women sees the empty tomb and tells the disciples about the empty tomb. She alone meets the angels and then Jesus meets her alone. Again, other ladies and Mary meet the angels. Jesus meets all of them on their way to the disciples.
So many events happened that day which are not contradictions but different details.
[PS: John himself says that the women (Mary Magdalene) went twice to the apostles' place (John 20:2 and John 20:18); first to report about the empty tomb and later to report about the risen Lord]