Officers said the driver hit the children at speed as he was driving away from another location close to the school, where his vehicle had brushed against a pedestrian.
The incident took place in Muzaffarpur district, around 43 miles (70 kilometres) from the state capital Patna.
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"Eyewitnesses have told us that the driver didn't stop even after running over these middle school children," Anil Kumar Singh, deputy inspector general of Bihar police told.
"His car overturned after the accident and he ran away from the spot. We are still looking for him," he said.
The 20 injured children have been admitted to hospital, he added.
India has some of the world's deadliest roads.
At least 12 children aged between seven and 14 were killed in northern Uttar Pradesh state last January after their schoolbus collided with a truck.
More than 150,000 people are killed each year with most accidents blamed on poor roads, badly maintained vehicles and reckless driving.
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