The question asks if an older (65+ year old) who is a 10th dan black belt can "defeat" a younger (say, 20 year old) 1st dan black belt.
I agree with the other answers here. But I just wanted to clarify one very important detail of the question that was asked.
What was asked is whether or not the older, higher ranked black belt can defeat the younger lower ranked black belt. The word, "defeat" implies fighting.
There is a big distinction between what is a "fighter" and what is a "martial artist". People often confuse the two. They are not the same.
A fighter is someone who has a good amount of experience actually fighting. Whereas a martial artist is someone who might have mastery of various martial arts skills, but who may or may not have experience actually fighting.
A fight happens when someone is trying to hurt you, and you have to stop that person.
Obviously, fighting on the street is fighting. But so is boxing and MMA. So is grappling when it's done to hurt you.
Sparring nicely, where you know you're not going to get that hurt, doesn't really count as fighting.
I love this video discussing the difference between a fighter and a martial artist:
Now, having clarified the difference between a martial artist and a fighter, let's ask the question now. Can a 65 year old 10th dan fight and win against a 20 year old 1st dan?
The 65 year old is at a big disadvantage. By that age, bodies generally (not always) are much weaker and more fragile than a body in peak condition. And age 20 is about the time when you're in peak physical condition. For many people, anyway. There are obviously exceptions to the rule.
Now, what the 65 year old 10th dan has is great skill. That's obvious by his 10th degree black belt. You generally don't get that rank without mastering technique. Yes, there are exceptions.
But is he a fighter? Most martial arts do not train people to be fighters. By age 65, most people have long stopped practicing for the purpose of getting better at actually fighting. They're in it to be martial artists, not fighters.
Whereas the 20 year old probably still has fighting as his central focus. But even the 20 year old may not be a fighter.
If the 65 year old used to be a fighter, and the 20 year has no fighting experience, then I might be tempted to put my money on the 65 year old. Experience at actually fighting is the key. That's probably going to make up for a lack of physical strength.
But if both the 65 year old and the 20 year old are legit fighters, then I'll bet on the 20 year old, especially if the 65 year old has not kept up his physical conditioning.
The extra degrees you have on your black belt generally are meaningless when it comes to fighting ability. Those stripes are there to represent that you've gained skill in the martial art. It doesn't say anything about your experience at fighting.
Hope that helps.