Counting ordered selection with repeats allowed

You are given a sack containing $n$ red balls and $n$ blue balls. You take the balls out of the sack one by one and write down the sequence of reds and blues you get. How many sequences are possible?

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HINT: A sequence is completely determined by the position in it of the $n$ red balls: once you know where they are, you also know where the blue balls are. How many different sets of $n$ positions are there in a string of $2n$ positions?