Everyone meets everyone once (matches or handshakes) among n people: how many

This type of question, asked 2 times: 12 Sep 2025 S3, 21 Sep 2025 S1. All 2 questions below, oldest first.

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Question 1
12 Sep 2025 S3
25 $In a tournament of 7 players, each player plays every other player once. How many matches are there?$
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Question 2
21 Sep 2025 S1
24 $In a room of 6 people, how many different handshakes are possible if each person shakes hands once with every other?$
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