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Step-by-Step Solution
Step 1: Understand the Problem
We need to form 5-digit numbers from the digits {0, 2, 3, 4, 7, 9} (repetition allowed), list them in ascending order, and determine the position (serial number) of the number 42923 in this ordered list. Since the number must be 5-digit, the first digit cannot be 0.
Step 2: Identify the Possible Leading Digits
The possible leading digits for a valid 5-digit number are {2, 3, 4, 7, 9}. We note that 0 cannot be used as the leading digit, but it can appear in other positions.
Step 3: Count All Numbers with Leading Digit Less Than 4
Digits less than 4 in the set {2, 3, 4, 7, 9} are 2 and 3. For each of these leading digits, the remaining 4 positions can be filled by any of the 6 digits {0, 2, 3, 4, 7, 9} with repetition allowed.
Number of such 5-digit numbers:
2 \times 6^4 = 2 \times 1296 = 2592
These are all the numbers that start with 2 or 3.
Step 4: Count Numbers Starting with 4 but Second Digit Less Than 2
Now consider 5-digit numbers starting with 4. The second digit can be {0, 2, 3, 4, 7, 9}, but we are interested only in those second digits less than 2, i.e., only 0.
For the second digit = 0, the remaining 3 positions can be any of the 6 digits:
1 \times 6^3 = 1 \times 216 = 216
Add this to our running total:
2592 + 216 = 2808
Step 5: Count Numbers Starting with 42 but Third Digit Less Than 9
The third digit belongs to {0, 2, 3, 4, 7, 9}. We want third digit less than 9, i.e., {0, 2, 3, 4, 7}, which is 5 possibilities. The remaining 2 positions can each be any of the 6 digits:
5 \times 6^2 = 5 \times 36 = 180
Add to our running total:
2808 + 180 = 2988
Step 6: Count Numbers Starting with 429 but Fourth Digit Less Than 2
The fourth digit is from {0, 2, 3, 4, 7, 9}. We want digits less than 2, i.e., only {0}, which is 1 possibility. The remaining 1 position can be any of the 6 digits:
1 \times 6^1 = 6
Update the total:
2988 + 6 = 2994
Step 7: Count Numbers Starting with 4292 but Fifth Digit Less Than 3
The possible fifth digits from {0, 2, 3, 4, 7, 9} that are less than 3 are {0, 2}, which is 2 possibilities. Each corresponds to a unique number since we are at the last position:
2
Add to the total:
2994 + 2 = 2996
Step 8: Determine the Rank of 42923
The total we have (2996) counts all the 5-digit numbers that come before 42923 in ascending order. Hence the serial number (or rank) of 42923 is:
2996 + 1 = 2997
Final Answer
The serial number of 42923 in the list is 2997.
Below is the referenced solution image for additional clarity:
