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The Apollo spacecraft, launched by Saturn V, the most powerful rocket ever build, could only take 3 man to the Moon. But how can SpaceX's Falcon Heavy, second to Saturn, take seven crew to Mars on a single Dragon V2?

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The Falcon Heavy cannot send humans to Mars in a single launch and probably won't ever send humans to Mars in any mission scenario. I believe the source of your confusion stems from the recent postings of Elon Musk on instagram. These 5 images depict the EDL of a Crew Dragon in a mission known as Red Dragon. Red Dragon is a mission architecture developed by NASA(not SpaceX) in which a Dragon capsule is sent to Mars as a large lander. Since Crew Dragon has 8 SuperDraco thrusters and a robust heatshield designed for Mars, it could land on Mars with a sizable payload if the fuel supply was increased.

Anyways your main question is why can Crew Dragon plus Falcon 9/Falcon Heavy seat 7 when Apollo plus Saturn V only sat 3. The answer is that the capsules were designed for different roles, different destinations, and in two different time periods. Crew Dragon is a LEO Taxi. It is designed to be fully automated and take people to and from the ISS. It has the internal cargo to seat 7 but will most likely fly with 4 astronauts and cargo.

Apollo is the crew module for a lunar mission. Its service module contains the consumables and propulsion equipment necessary to go to the Moon. The mission plan called for 3 astronauts so it was designed with 3 seats. There was actually a plan to have a 5 person Apollo capsule in order to rescue a stranded Skylab crew.
Apollo Rescue Capsule

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