1. **State the problem:** We want to find how many 5-letter arrangements can be made from the 26 letters of the English alphabet with no repeated letters. 2. **Formula used:** The number of permutations of $k$ objects chosen from $n$ distinct objects without repetition is given by: $$P(n,k) = \frac{n!}{(n-k)!}$$ 3. **Apply the formula:** Here, $n=26$ (letters in the alphabet) and $k=5$ (letters chosen). $$P(26,5) = \frac{26!}{(26-5)!} = \frac{26!}{21!}$$ 4. **Simplify the factorial expression:** $$\frac{26!}{21!} = 26 \times 25 \times 24 \times 23 \times 22$$ 5. **Calculate the product:** $$26 \times 25 = 650$$ $$650 \times 24 = 15600$$ $$15600 \times 23 = 358800$$ $$358800 \times 22 = 7893600$$ 6. **Final answer:** There are **7,893,600** different 5-letter arrangements possible with no repeated letters from the English alphabet.