GingerAle

Tthe number of ways of choosing three people in a circular table with 12 people, so that no two people are next to each other.

Solution:

There are 12 ways to choose 3 persons who are adjacent (next to each other).

There are 12 ways to choose 2 persons who are adjacent.

There are 8 ways to choose 1 person who is not adjacent to the other two (2).

There are nCr(12,3) to choose a set of three persons from set of 12.

Subtract the persons who are adjacent from the total sets.

$\text{So then ... } \dbinom {12}{3} - 12 - (12*8) = 112\\ \small \text{ There are 112 ways to select three persons from a set of 12 seated in a circle, such that no two persons are adjacent. }\\$

GA

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Max, your monographic summary analysis on binomial theory is amazing! It’s one of the best I’ve read anywhere. 

It’s wonderful you have returned! I hope you stay around for awhile.

I will let you figure out the rest.

I don’t know if the OP student will choose to figure out the rest, but I sure as hell will.

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You may find this formula interesting.

$\sum \limits_{k=0}^{m}\binom{n}{k} (-1)^k \binom{p – s*k – 1}{p – s*k – n} \leftarrow \text {where p is the point (sum) target, n is the # of die, }\\ \hspace {47mm} \text{s is the number of sides, and } m=\lfloor \dfrac{p-n}{s}\rfloor \\$

This formula is from J. V. Uspensky’s Introduction to Mathematical Probability (1937).

This calculates the number of combinations for any given sum of pips on (N) number of dice with (S) sides, where each die-side has a unique number of pips from 1 to (S)

The reason for the loss of points and missing posts is on or about February 29 or March 1, 2024 the web2.0calc databases (and maybe the operating systems) were restored to January 16, 2024. The German forum was restored to December 8, 2023.  All posts (and related point tallies) subsequent to the restore point and prior to the restore time no longer exist on the forum.

I can only speculate on the reasons for restoring to a point six weeks prior: A corrupted database index is the most likely cause. These can be difficult and time consuming to repair if the errors are nonlinear or compounded over a large span of time.   

I wonder if it is due to the amount of repeat questions (and most likely homework questions) being posted in this time period? Anyone have any answers???

The vast majority of questions posted on the forum are posted by scripted bots, under the control of a few psychopathic and obsessed trolls. Some of the answers are also posted by scripted bots; these answers may be generated by AI, and at least 20% are incorrect, with wrong or missing procedures and/or intentional math errors. There are a few legitimate questions posted by humans. There are also bot-posted questions that imitate   

This forum is a real microcosm of the science fiction narratives where the world is invaded and takeover by human-like cybernetic virtual beings under the control of biological humans. It’s bloody creepy!!!

GA

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