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xkcd #2909: Moon Landing Mission Profiles
(imgs.xkcd.com)
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If you pick a low enough orbit, it gives you a lot of freedom to use a lightweight launch vehicle such as a stepladder.
Because the main goal was to beat the Russians to the moon. They also came up with a plan to fly astronauts to the moon then later devise a way to get them back.
I heard they would have needed a bigger rocket for that, in order to account for giant clanking brass ones adding to the launch weight.