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Step-by-Step Solution
Step 1: Understand the Task
We want to list all six-digit numbers formed by choosing digits x_1, x_2, x_3, x_4, x_5, x_6 under the condition 0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6 . The digits must be strictly increasing from left to right. We then order all such 6-digit numbers in ascending order and need to find the sum of the digits of the 72nd number in this list.
Step 2: Count Numbers Beginning with 1
If x_1 = 1 , we must choose the next five digits ( x_2, x_3, x_4, x_5, x_6 ) from the remaining digits 2 to 9. The number of ways to choose 5 distinct digits out of 8 (2 through 9) is
{8 \choose 5} = 56.
Thus, the first 56 numbers in our list will start with 1.
Step 3: Identify When x_1=2
Once we exhaust numbers starting with 1, we move to numbers where x_1 = 2 . The next digits x_2, x_3, x_4, x_5, x_6 will be chosen from 3 to 9. The number of ways to choose 5 digits from the 7 available (3 through 9) is
{7 \choose 5} = 21.
This means the signs for these numbers run from the 57th up to the 56 + 21 = 77th position in the overall list.
Step 4: Locate the 72nd Number
Since the first 56 numbers start with 1, the 72nd number lies among those starting with 2. Within the block where x_1=2 , we need the (72 − 56) = 16th number (because numbers 57 through 77 in the entire list correspond to the 1st through the 21st when x_1=2 ).
Step 5: Find the 16th Combination When x_1=2
We list (in ascending order) all 5-element combinations from the digits {3, 4, 5, 6, 7, 8, 9} and pick the 16th one:
3,4,5,6,7
3,4,5,6,8
3,4,5,6,9
3,4,5,7,8
3,4,5,7,9
3,4,5,8,9
3,4,6,7,8
3,4,6,7,9
3,4,6,8,9
3,4,7,8,9
3,5,6,7,8
3,5,6,7,9
3,5,6,8,9
3,5,7,8,9
3,6,7,8,9
4,5,6,7,8
The 16th combination is (4, 5, 6, 7, 8). Therefore, the 72nd number overall is 245678 .
Step 6: Sum the Digits of the 72nd Number
The digits of 245678 are 2, 4, 5, 6, 7, and 8. Their sum is:
2 + 4 + 5 + 6 + 7 + 8 = 32.
Hence, the sum of the digits in the 72nd number is \boxed{32} .
