1. **State the problem:** Solve for $r$ in the equation $$10 P_{n,r} = 10 P_{n,r+1}$$ where $P_{n,r}$ is the number of permutations of $n$ items taken $r$ at a time. 2. **Recall the formula for permutations:** $$P_{n,r} = \frac{n!}{(n-r)!}$$ 3. **Substitute the formula into the equation:** $$10 \cdot \frac{n!}{(n-r)!} = 10 \cdot \frac{n!}{(n-(r+1))!}$$ 4. **Cancel the common factor 10:** $$\cancel{10} \cdot \frac{n!}{(n-r)!} = \cancel{10} \cdot \frac{n!}{(n-r-1)!}$$ 5. **Cancel the common factor $n!$ on both sides:** $$\frac{\cancel{n!}}{(n-r)!} = \frac{\cancel{n!}}{(n-r-1)!}$$ 6. **Rewrite factorials to relate denominators:** $$(n-r)! = (n-r)(n-r-1)!$$ 7. **Substitute into the left side denominator:** $$\frac{1}{(n-r)(n-r-1)!} = \frac{1}{(n-r-1)!}$$ 8. **Multiply both sides by $(n-r)(n-r-1)!$ to clear denominators:** $$1 = (n-r)$$ 9. **Solve for $r$:** $$r = n - 1$$ **Final answer:** $$\boxed{r = n - 1}$$