Puzzle: Pick Up Lines

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Arriving late at a party and after greeting a few people, I found myself standing at the entrance next to a clever and stylish mathematician named Caroline. Skipping any form of greeting, she said, "There are nine people including us at this party. Could each person at this party have shaken hands with a different number of people, given that germophobe Ron over there hasn't shaken hands with anyone?"

I took a while before answering.

Warm-Up:

How did I answer?

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Solution:
If this were the case, then the number of hands shaken by each person would be:

0, 1, 2, 3, 4, 5, 6, 7, 8

But if someone shook hands with 8 people, then Ron (who has shaken hands with nobody according to Caroline) would have had to shake hands with that person. This cannot be.

Problems:

1. "Not bad," said Caroline, an impish smile crossing her lips. "In fact, only two people have shaken hands with the same number of people. How many hands has each of that pair shaken?"

As a token of recognition of my answer, Caroline deigned to shake my hand. Smooth and cool to the touch, very nice. She didn't allow her hand to linger however. "This raised both our counts by one," she observed. "I have shaken hands with more people than you this evening and you should be able to figure out how many that is. If you do, I'll fetch you a drink."

2. How many people's hands (including mine) had Caroline shaken?

"Orange juice, please" I said to my new friend after I told her my solution, "with a touch of vodka."

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