We need the serial number of the 5-digit number $42923$ when all 5-digit numbers formed from digits $\{0,2,3,4,7,9\}$, with repetition allowed, are arranged in ascending order.

Since it is a 5-digit number, the first digit cannot be $0$.

1. Allowed digits

Available digits are: $\{0,2,3,4,7,9\}$ So each position has $6$ choices, except the first position which cannot be $0$.

2. Strategy

To find the serial number of $42923$, count how many valid 5-digit numbers come before it in ascending order, then add $1$.

We compare digit by digit.

Number: $42923$

3. Count numbers with first digit less than $4$

Possible first digits for a 5-digit number are: $2,3,4,7,9$

Digits less than $4$ are $2,3$, i.e. $2$ choices.

For each such choice, the remaining 4 digits can be anything from the 6 allowed digits: $6^4 = 1296$

So count: $2 \times 6^4 = 2 \times 1296 = 2592$

4. Fix first digit $4$, count numbers with second digit less than $2$

Allowed digits less than $2$ from the set are only: $0$

So there is $1$ choice for second digit.

Remaining 3 digits can be anything: $6^3 = 216$

So count: $1 \times 216 = 216$

Running total: $2592 + 216 = 2808$

5. Fix first two digits $42$, count numbers with third digit less than $9$

Allowed digits less than $9$ are: $0,2,3,4,7$

That is $5$ choices.

Remaining 2 digits can be anything: $6^2 = 36$

So count: $5 \times 36 = 180$

Running total: $2808 + 180 = 2988$

6. Fix first three digits $429$, count numbers with fourth digit less than $2$

Allowed digits less than $2$ are only: $0$

So count: $1 \times 6 = 6$

Running total: $2988 + 6 = 2994$

7. Fix first four digits $4292$, count numbers with fifth digit less than $3$

Allowed digits less than $3$ are: $0,2$

That is $2$ choices.

So count: $2$

Running total: $2994 + 2 = 2996$

8. Add 1 for the number itself

Therefore serial number of $42923$ is: $2996 + 1 = 2997$

9. Comparison with stored answer

Derived answer: $2997$

Stored correct answer: $2997$

They match.