Explanation:

Since the year of birth is known, the birthday being on Saturday can have a zero probability.

Also since between 6th

 and 10th, there are three days 
i.e. 7th,8th and 9th

As pointed out by Ankit in the comments, we have following possibilities on these three dates,

1. Monday, Tuesday, Wednesday
2. Tuesday, Wednesday, Thursday
3. Wednesday, Thursday, Friday
4. Thursday, Friday, Saturday,
5. Friday, Saturday, Sunday
6. Saturday, Sunday, Monday
7. Sunday, Monday, Tuesday

Out of these 7 events, we have 3 chances of his birthday falling on saturday.

⇒Probability= FavorableEvents /TotalEvents

⇒Probability=3/7

Therefore, the probability of birthday falling on Saturday can be 3/7.

Hence, optio 1 is the correct answer.