KANSAS CITY, Mo. — Oregon coach Dana Altman made his way slowly to each player’s locker Saturday evening, sipping water from a plastic cup. He looked less elated than pleasantly content, like a man about to settle down in the good chair after a hard day’s work.

The only thing that betrayed the euphoria and improbability of the moment was his shirt. It was all wet. The Ducks, after reaching the school’s first Final Four since 1939, had dumped a bucket of ice over their aww-shucksing coach.

Altman took it, “as uncomfortable as all get out,” assistant coach Mike Mennenga said.

A similar scene played out across the country Saturday and into Sunday as wide-eyed newcomers, Gonzaga and South Carolina, joined Oregon and powerhouse North Carolina in the Final Four.

In San Jose, Calif., Gonzaga, author of so many NCAA tournament upsets, wrote one of its biggest yet: The Bulldogs had inspired their gentle, reserved coach, Mark Few, to jump into the arms of his 300-pound center, Przemek Karnowski.

“Why not?” he said, describing the theme of the evening.

In New York, South Carolina coach Frank Martin, he of the permanent dad-picking-you-up-from-detention glare, was near tears.

He, too, asked his team the question of the weekend: “Why not?”

Why not make it to the Final Four? For these three teams, the reasons were plenty.

Oregon’s last Final Four was the first one ever. The Ducks lost a key player, forward Chris Boucher, right before the tournament started but persevered.

Gonzaga was once a cute story, the small Jesuit school from the Northwest that reached the regional finals in 1999 and inspired Gus Johnson’s immortal call, “The slipper still fits!” The slipper went dancing into the tournament for the 19th season in a row this year but still had never made it to the Final Four.

Which was better than South Carolina. The Gamecocks hadn’t won a single NCAA tournament game since 1973.

But this year’s Final Four has become one for grinders, three less-than-storied programs who have performed in the image of the work-a-day basketball lifers who coach them.

And then there is mighty North Carolina, which has now made more Final Four appearances, 20, than any other team after surviving Kentucky in a classic on Sunday. North Carolina adds the star power, but this year’s collection of teams is an ode to quiet toil, and the career-defining joy for three coaches was hard to miss.

Altman, who grew up in tiny Crete, Neb., got his start as the coach and physical education instructor at the vaguely named Southeast Community College. He never thought he’d coach Division-I basketball, much less this.

“Today is just about our players,” Altman demurred, when asked about his journey to the Final Four. “I feel so good for them.”

He was trying hard to suppress a smile.

Few is the son of a small-town pastor from Oregon. He spent his entire career, from graduate assistant to head coach, at one school.

“I’ve been there 28 years,” he said Saturday. “My first year on staff we won four Division I games. And, I mean, this wasn’t even possible.”

The winning has shown unlikely sides of some of the coaches. Martin — son of a single mother from Cuba, a former Miami bouncer who has been shot at, the most intimidating man in the sport — was unmasked as a mush.

Earlier in the week, he engaged in a tender moment when a 13-year-old reporter asked him a question. He cried when South Carolina reached the regional semifinals, and he hugged his crying mother when they won in the finals.

“Just — I’m just out of words,” he said. “Out of words.”

But Roy Williams, himself a former junior varsity player at North Carolina who later had to supplement his coaching salary selling calendars, might be leading the favorite.

The question will be whether North Carolina’s talented frontcourt can figure out Oregon’s Jordan Bell, who blocked eight shots against Kansas and has been the most dominant defensive force of the tournament.

Gonzaga has been the most consistent team all season. The Bulldogs have lost only one game. But South Carolina brings one of the nation’s best defenses. The Gamecocks held Baylor to 50 points in the regional semifinal. They also stormed past one of the hottest teams anywhere, No. 2 Duke, in the second round.

A No. 7 seed, South Carolina was picked to lose in the first round by most brackets on ESPN.com. Three of every 1,000 brackets had the Gamecocks in the Final Four. But they had been dreaming of the opportunity since last season, when South Carolina was told by the NCAA that it had made the tournament … only to later say that there had been a mistake.

This year, said guard Sindarius Thornwell, “All we wanted was a chance.”

Now four improbable teams do.

Standing atop a ladder, Few shrugged his shoulders and said now that Gonzaga had made it this far, “Might as well win it.”

Why not?