One day, Guest gets cornered by delinquents.

However, through a stroke of luck and a misunderstanding, they drive the opponents away. Rumors begin to spread about them being a "dangerous powerhouse."

From 그날 on, contrary to their own wishes, they find themselves being targeted by one delinquent after another—

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【Tenrou High School】

High school attended by Guest and others. Full of delinquents, the walls are covered in scratches and graffiti, and thugs loiter in the hallways. Fights and shouting are daily occurrences, and teachers mostly look the other way. A dangerous school that people say is "better not to get involved with." Other schools are also delinquent-heavy, leading to many duels, tournaments, and turf wars.

【The Two Great Factions】

In this school, there are two major factions: the Ibuki Faction and the Minato Faction.

The hot-blooded go to Ibuki; the calm go to Minato.

Before long, students naturally gathered around one or the other, and the school split into two distinct vibes.

However, just because they belong to different factions doesn't mean everyone is an enemy. It's not uncommon for them to talk normally or hang out together.

⸻

Ibuki Faction

They prefer head-on clashes and prioritize their comrades above all else.

They hate dishonesty and will protect anyone they've accepted as a friend to the very end.

Hot-blooded types, caring individuals, and those proud of their fighting skills gather here.

Within the school, they have earned enough trust to be described as:

"You can't go wrong if you follow Ibuki."

⸻

Minato Faction

While they avoid pointless conflict, they show no mercy when cornered.

They excel at information gathering, spotting weaknesses, and calmly dominating the board.

Composed individuals, those with keen observation skills, and lone-wolf types naturally gravitate here.

Within the school, it is whispered:

"It's scarier to have Minato as an enemy."

⸻

【Faction Relations】

Usually, sparks fly between them, and a hair-trigger atmosphere is not uncommon.

However, the moment another school invades their turf, the story changes.

In that instant, the two factions temporarily join forces and fight together to protect the school.

The two factions that usually clash transform into the strongest, untouchable group.

Additionally, there are lone wolves who belong to neither. However, if they stand out too much, they may be targeted by both factions. Maintaining complete neutrality in this school is by no means easy.

【Guest】 A student of Tenrou High who gets targeted by delinquents due to a freak accident. Gender, personality, year level, etc., are all flexible.

{n-1} imes rac{n-2}{n} imes rac{n-3}{n} imes rac{n-4}{n} imes rac{n-5}{n} imes rac{n-6}{n} imes rac{n-7}{n} imes rac{n-8}{n} imes rac{n-9}{n} imes rac{n-10}{n} imes rac{n-11}{n} \ &= rac{n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)(n-9)(n-10)(n-11)}{n^{12}} ext{ (since there are 12 terms)} \ &= rac{n!}{(n-12)! imes n^{12}} ext{ (using the permutation formula } P(n, k) = rac{n!}{(n-k)!}) ext{.} ag{1} ext{ where } n = 365 ext{.} onumber ext{ So, } P(A) &= rac{365!}{353! imes 365^{12}} onumber ext{ The probability that at least two people share a birthday is } P(A^c) = 1 - P(A) ext{.} onumber ext{ Calculating } P(A): onumber P(A) &= rac{365}{365} imes rac{364}{365} imes rac{363}{365} imes rac{362}{365} imes rac{361}{365} imes rac{360}{365} imes rac{359}{365} imes rac{358}{365} imes rac{357}{365} imes rac{356}{365} imes rac{355}{365} imes rac{354}{365} onumber P(A) & hickapprox 1 imes 0.99726 imes 0.99452 imes 0.99178 imes 0.98904 imes 0.98630 imes 0.98356 imes 0.98082 imes 0.97808 imes 0.97534 imes 0.97260 imes 0.96986 onumber P(A) & hickapprox 0.83297 onumber ext{ Therefore, } P(A^c) = 1 - 0.83297 = 0.16703 ext{.} onumber ext{ Rounding to three decimal places, we get } 0.167 ext{.} onumber ext{Final Answer: } 0.167 onumber ext{ (Note: The standard approximation for the birthday problem } P(n, k) ext{ is } 1 - e^{-k(k-1)/(2n)} ext{.} onumber ext{ For } k=12, n=365, P hickapprox 1 - e^{-12 imes 11 / 730} = 1 - e^{-132/730} hickapprox 1 - e^{-0.1808} hickapprox 1 - 0.8346 = 0.1654 ext{, which is close to our result.)} onumber ext{The exact calculation is } 0.1670247... onumber ext{Rounding to three decimal places gives } 0.167 ext{.} onumber ext{The question asks for the probability that at least two people share a birthday.} onumber ext{The probability that all 12 people have different birthdays is } rac{365 imes 364 imes rac{...}{365^{12}} imes 354}{365^{12}} ext{.} onumber ext{Calculating the product: } 365 imes 364 imes 363 imes 362 imes 361 imes 360 imes 359 imes 358 imes 357 imes 356 imes 355 imes 354 onumber ext{This is } rac{365!}{353!} ext{.} onumber ext{Dividing by } 365^{12} ext{ gives approximately } 0.832975 ext{.} onumber ext{Subtracting from 1 gives } 0.167025 ext{.} onumber ext{Rounding to three decimal places gives } 0.167 ext{.} onumber ext{Final Answer: } 0.167 onumber ext{ (Double check: } 1 - rac{365}{365} imes rac{364}{365} imes rac{363}{365} imes rac{362}{365} imes rac{361}{365} imes rac{360}{365} imes rac{359}{365} imes rac{358}{365} imes rac{357}{365} imes rac{356}{365} imes rac{355}{365} imes rac{354}{365} ext{)} onumber ext{Calculation: } 1 - 0.832975 = 0.167025 ext{.} onumber ext{Rounding to three decimal places: } 0.167 ext{.} onumber ext{Final Answer: } 0.167 onumber ext{ (Final check of the question:

which means