The Arrow Stage Lines bus was southbound on State Route 163 when it failed to negotiate a curve, went off the road and rolled over several times at 7:30 p.m. Sunday, said Trooper Cameron Roden of the Utah Highway Patrol.
"When the vehicle was overturning, the roof of the bus split open and multiple occupants of the vehicle were ejected," he said.
The driver suffered minor injuries, he said.
A woman who answered the phone at Arrow Stage Lines headquarters in Omaha, Neb., early Monday said no one would be available to answer questions until later in the morning.
Roden said the bus was en route from Telluride, Colo., to Phoenix after a ski trip.
A manifest showed 51 passengers were on board the bus when it crashed about 10 miles north of Mexican Hat, in the Four Corners region where Utah meets Arizona, Colorado and New Mexico. Rescue crews from all four states were sent to help.
Trooper Cameron Roden told CBS station KUTV that slippery road conditions may have contributed to the crash.
"They say there had been light rain in the area. They say it doesn't appear it was snowing in the area, but they are looking at every aspect right now," said Roden.
The region's isolation and the winter weather slowed aid efforts Sunday. A helicopter from Colorado could not take off because of winter storms. Cell phone service was not available in the area, Roden said.
Twenty of the injured were taken to San Juan Hospital in Monticello, 80 miles north, said hospital chief executive Craig Preston. Ten to 13 still were being evaluated while others with more serious injuries were sent elsewhere.
"We had some pretty severe stuff," Preston said.
Another bus was sent to pick up passengers who weren't badly injured.
Elsewhere, two people were killed Monday morning in a car that collided with a school bus near Bartow, Fla., the Polk County sheriff's office said. None of the 11 students on the bus or their driver were injured. Deputies said the car veered into the path of the bus.
In Mapleton, Maine, a school bus went out of control at a sharp turn, clipped a utility poll and landed on its side, sending eight people to a hospital with non-life threatening injuries Monday morning, police said.
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