1. **Problem statement:** We want to find the number of ways to arrange twelve different amino acids into a polypeptide chain of length five. 2. **Formula used:** Since the order matters and we are selecting 5 amino acids out of 12 without repetition, this is a permutation problem. The number of permutations of $k$ items from $n$ distinct items is given by: $$P(n,k) = \frac{n!}{(n-k)!}$$ 3. **Explanation:** Here, $n=12$ (total amino acids) and $k=5$ (length of the chain). We calculate: $$P(12,5) = \frac{12!}{(12-5)!} = \frac{12!}{7!}$$ 4. **Intermediate work:** $$12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7!$$ So, $$P(12,5) = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{7!} = 12 \times 11 \times 10 \times 9 \times 8$$ 5. **Calculation:** $$12 \times 11 = 132$$ $$132 \times 10 = 1320$$ $$1320 \times 9 = 11880$$ $$11880 \times 8 = 95040$$ 6. **Final answer:** There are **95040** different ways to arrange twelve amino acids into a polypeptide chain of five amino acids.