1. **Problem:** Find the number of permutations of 7 different objects taken 4 at a time without repetition. 2. **Formula:** The number of permutations without repetition is given by $$P(n,r) = \frac{n!}{(n-r)!}$$ where $n$ is the total number of objects and $r$ is the number taken at a time. 3. **Calculation:** Here, $n=7$ and $r=4$. $$P(7,4) = \frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{7 \times 6 \times 5 \times 4 \times 3!}{3!} = 7 \times 6 \times 5 \times 4 = 840$$ 4. **Explanation:** We calculate the factorial of 7 and divide by the factorial of the difference (3) to avoid counting repeated arrangements. **Final answer:** The number of permutations is **840**.