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choose(n, k) := product(k + 1 ... n) / product(1 ... n - k);
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choose(n,k) := binomial(n, k, 1, 1);
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binomial(n, k, i, result) :=
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result when i > k else
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binomial(n, k, i + 1, (result * (n - i + 1)) / i);
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