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All swimming strokes take time to be perfected. These descriptions may be able to help you out with some of the fine details.
Backstroke
Backstroke must be swum on the back at all times. Technically, in backstroke events, any combination of arm and leg movements can be used as long as the swimmer stays on their back. However, the stroke used in competitive swimming is the back crawl, which is what I'm describing here.
The Stroke
Keep your body as flat as possible at all times.
Your arms will give you most of your power. As one arm is pulling underwater, the other one is recovering (this gives a windmill effect).
You want to enter your arm into the water with your little finger first. Then pull our arm back toward you. Your elbow will bend after about 10 inches or so into this, which will cause your hand to apply force. Your arm will end up making a "S" or "Question Mark" shape during this process.
The kick is just a normal flutter kick, with both legs kicking alternately. Legs should be kept fairly straight, and you will get about 6 kicks for every complete arm cycle you do.