1. The problem is to solve the permutation equation $P(n,7) = 8,648,640$ where $P(n,7)$ represents the number of permutations of $n$ items taken 7 at a time. 2. The formula for permutations is: $$P(n,k) = \frac{n!}{(n-k)!}$$ where $n!$ is the factorial of $n$ and $k$ is the number of items chosen. 3. Here, $k=7$, so: $$P(n,7) = \frac{n!}{(n-7)!} = 8,648,640$$ 4. We need to find $n$ such that: $$\frac{n!}{(n-7)!} = 8,648,640$$ 5. This expands to: $$n \times (n-1) \times (n-2) \times (n-3) \times (n-4) \times (n-5) \times (n-6) = 8,648,640$$ 6. We try integer values for $n$ starting from 7 upwards: - For $n=10$: $$10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 = 604,800$$ (too small) - For $n=11$: $$11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 = 3,326,400$$ (too small) - For $n=12$: $$12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 = 19,958,400$$ (too large) 7. Check $n=11$ and $n=12$ more carefully: - $n=11$ product is $3,326,400$ - $n=12$ product is $19,958,400$ 8. Since $8,648,640$ lies between these two, try $n=11$ with one more factor: $$11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 = 3,326,400$$ 9. Try $n=13$: $$13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 = 62,748,480$$ (too large) 10. Try $n=9$: $$9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 = 181,440$$ (too small) 11. Try $n=14$: $$14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 = 87,178,291,200$$ (too large) 12. Since direct trial is tedious, factorize $8,648,640$ to find $n$: 13. Factor $8,648,640$ into prime factors: $$8,648,640 = 2^7 \times 3^3 \times 5 \times 7 \times 11 \times 13$$ 14. Notice the factors correspond to consecutive integers from 7 to 13: $$7 \times 8 \times 9 \times 10 \times 11 \times 12 \times 13 = 8,648,640$$ 15. Therefore, $n=13$ satisfies: $$P(13,7) = 8,648,640$$ **Final answer:** $$\boxed{13}$$