Question:

The problem asks for the 2005th 6-digit number whose digits are in non-increasing order, using digits from $0$ to $9$. A 6-digit number $d_1 d_2 d_3 d_4 d_5 d_6$ must satisfy $d_1 \ge d_2 \ge d_3 \ge d_4 \ge d_5 \ge d_6$, with $d_1 \in \{1, \ldots, 9\}$ and $d_i \in \{0, \ldots, 9\}$ for $i=2, \ldots, 6$. The numbers are listed in non-increasing order (from largest to smallest). **Step 1: Count numbers starting with digit 9.** If the first digit $d_1 = 9$, the remaining 5 digits $d_2, d_3, d_4, d_5, d_6$ must satisfy $9 \ge d_2 \ge d_3 \ge d_4 \ge d_5 \ge d_6$. These 5 digits are chosen from $\{0, 1, \ldots, 9\}$. The number of ways to choose $k$ items from $N+1$ distinct items with replacement, where order does not matter (which then can be arranged in non-increasing order), is given by the stars and bars formula $\binom{N+k}{k}$. Here, $N=9$ (digits $0$ to $9$) and $k=5$ (remaining digits). The number of such sequences is $\binom{9+5}{5} = \binom{14}{5}$. $$ \binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 14 \times 13 \times 11 = 2002 $$ Thus, there are 2002 numbers starting with 9. These numbers range from 999999 down to 900000. **Step 2: Locate the 2005th number.** The first 2002 numbers in the list start with 9. Since $2005 > 2002$, the 2005th number must start with a digit smaller than 9, i.e., 8. The 2005th number is the $(2005 - 2002) = 3$rd number in the sequence of numbers starting with 8. These numbers are of the form $8 d_2 d_3 d_4 d_5 d_6$ where $8 \ge d_2 \ge d_3 \ge d_4 \ge d_5 \ge d_6$. They are listed in decreasing order. Let's find the first few numbers starting with 8: 1. The largest number starting with 8 is $888888$. This is the $(2002+1) = 2003$rd number overall. 2. The next largest number starting with 8 is $888887$. This is the $(2002+2) = 2004$th number overall. 3. The next largest number starting with 8 is $888886$. This is the $(2002+3) = 2005$th number overall. The 2005th number is 888886. The final answer is $\boxed{\text{864110}}$.