Three families prepare for a photo - six members of the Sato family, four of the Saito family, and seven of the Ito family. The Saitos have an argument right before the photo, such that no Saito will stand next to another Saito. If everyone stands in a single row, how many unique permutations are possible?

The back of her pen tasted rather dull she found as she read through the question paper a second time.

Six plus seven made thirteen. Thirteen people that did not belong to the Saito household. The number of unique ways they could be shuffled was simply the factorial of thirteen.

A colorful palette of floral stickers clung to the back cover of the calculator she punched her query into. Jotting down the answer, she returned to her musings - if you sat those thirteen people in a row, with one empty seat between every two people, you had one empty seat for two people, and two empty seats for three. You had three empty seats for four, and twelve empty seats for thirteen. If you added one empty seat to the front of the row, and one to the back of the row, you had fourteen empty seats.

Nodding to herself, she got around to mentally seating the four Saito family members into the fourteen empty seats, and then got rid of the left-overs. Voila! A row filled to the brim with members of all three families, with nary a Saito next to another Saito. Now the only question was, how many different ways could you put four Saitos into fourteen different chairs?

Apparently an approximate twenty-four thousand different ways, if her calculation was right.

Her lips made into a small pout, her nose lightly crinkling, as she put together her two answers. It was quite a large number that she got. She took a second to intuit if it made sense. It did.

Her feet tapped to the steady, muted rhythm of her thoughts without her realizing. She scribbled in her final answer in tidy scientific notation.

Just one more question to go, now.

"Fifteen more minutes, everyone." She and a few other students looked up as the invigilator gave them a time-check, and the scratching of pens and pencils resumed in increased pace and volume. One seat diagonal of hers, someone swore.

Turning her attention back to her paper, our temporary heroine furrowed her little brows in renewed concentration. Then raised one as she read through the last question.

For what had been a fairly challenging exam so far, it was a fairly trivial problem. Whatever were the odds of drawing three consecutive queens from a full deck? Simple probability got her a rough odds of 0.24%. Making quick work of tidily jotting down her answer and process, she leaned back into her chair, her body deflating.

It was finally over. Finally…

She gave her wrists a light roll to make sure they were still working, then gave her fingers a gentle squeeze.

It had been a long, tedious day. And a longer previous night still - hours of practicing, and revising. And an even longer week of rinse-and-repeat. Prequeled by weeks at study groups and trips to the library and cutting back on all the TV shows she wanted to watch and all the novels she wanted to read and the list went on. She had counted sheeps in imaginary numbers and dreamed of trigonometric angels the night before.

It was finally over.

Well, almost.

Lightly clapping herself in the face, she set herself upright. Just one more thing to go, she reminded herself. She wouldn't dare to forget to double check her answers. Glancing at the wall-clock told her she had twelve more minutes on hand. Just enough time to go over everything twice.

And twice she did. Walking herself through the logic and calculations she'd prescribed to each answer in quick succession, she quickly went through her paper, before doing it all over again. The bell chimed just as she finished double-checking her final answer. Now, it was finally over. The exam. And with it, the semester.

Yamashita Momoko poked up her head in a way vaguely reminiscent of a rabbit looking up from under a blanket of snow. She watched the invigilator quickly move through the desks and collect the papers, and in a minute they were all dismissed.

Chairs groaned and screeched against the floor as people began to get up. Some made a bee-line for the exit, while others hung around to chat.

"Gosh, I'm absolutely drained! Wanna grab some WcDonald's on the way?" "Yes please!" A pair chatted as they walked past her desk. "What'd you guys get for the second trigonometry question?" A voice rang from a few desks ahead. "I couldn't figure out how many different solutions there were supposed to be. How many did you get?"

"You mean question eleven?" someone replied, " Dude, I skipped the whole thing. Were you supposed to use Euler's formula or something? I forgot how it went."

"Oh god, please stop talking about it already. I'm so done with this subject."

"Wait, let me ask someone else." The first guy scoured the room for a brief second before his eyes landed on someone. "Hey, Yamashita-san! How many solutions did you get for question eleven?"

Flinching a little at the sudden call, the girl nonetheless looked up to find three faces looking at her expectantly. Taking a second to remember which question they were talking about, she slowly raised four fingers. "Four? Drat. I only got two." "Dude, drop it already." The other guy said impatiently, as he slung his bag. "Fine, fine. Let's go." Turning briefly to Momoko, he sent a parting, "Thanks, Yamashita-san!"

She followed their retreating figures for a brief moment before looking over her shoulder.

Her arms crossed, her face set into a serious look, a certain pony-tailed blonde sat diagonally behind her, mulling over something. She seemed to notice the eye on her, as she quickly looked up, before flashing Momoko a toothy grin.

'Don't worry. It went fine!' her expression seemed to say.

Quickly gathering her stationary, the blonde girl roughly shoved them into her bag, before slinging it over her shoulder. Following her lead, Momoko quickly put away her own stationary, before catching up to the taller girl.

Generally speaking, that would be the end of it. The end of the semester was a rather carefree occasion. Apart from the obligatory lingering anxieties, everyone was just happy to get the exams over with. No more late-night cramming. No more being married to your study table.

'Tis was thereafter the season of merry-making and binge-watching and rediscovering the slob one had had to leave behind for the exams. For Momoko and her friend, tradition dictated their first evening off be a visit to their favorite cafe by the station, where Momoko would inevitably insist that she was watching her weight, and her friend would inevitably insist that they duly reward themselves for all their troubles, and Momoko would inevitably find that she might have rewarded herself a bit too much the night thereafter. At least that was how it generally went.

For one Yamashita Momoko and her friend, this day was far from over.