You have found how many individual pairs can be made from 8 people.
That is NOT the question.
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OK, I may have misinterpreted the question. You explained yours and Alan’s focus clearly.
My focus is on this part: “. . . In how many ways could the people be paired?” This is where the question mark is. However, if the sequence of these unique pairs entering a room is a consideration then my solution is obviously incomplete.
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If there is three people, or any odd number of people in a room, they cannot all be paired of so the question would make no sense. There would be no answer.
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Actually, they can, just not all of them simultaneously when the count is odd.
Reiteration: This formula gives the unique combinations of pairs (sets of 2) from a set of {n}.
It’s rather interesting why this works for both odd and even numbers. This sequence is an offset of Gauss’ “Sum of Sequential Numbers” formula. However, in this case, the formula adds, not the first N numbers, but the first (N-1) numbers, aligning it with the unique combinations of 2 from set {n}. (This does not have a unique OEIS ID).
Here's why even and odd counts solutions use the same formula in Gauss’ “Sum of Sequential Numbers.”
For an even count:
\( \text{Number of pairs * Sum of each pair} \large = (\frac{n}{2})(n+1) = \frac{n(n+1)}{2}\)
For an odd count
\(\text{Number of pairs * Sum of each pair} = \Large (\frac{n + 1}{2})(n) = \frac{n(n+1)}{2} \)
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+1038 