More than likely the problem is with the transmission range switch. This is a multifunction switch. It tells the computer what gear the transmission is in, it turns on the back-up lights, and it prevents the starter from operating unless the transmission is in park or neutral.
As mentioned, the next time the engine won’t start leave the ignition switch on so the dash lights are on. Step on the brake pedal and shift the transmission into neutral and then try starting the engine. If the engine starts then the problem is with the transmission range switch.
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