Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and ar

Q: Wait, why can't the first digit be zero? Isn't zero a digit?

That is a great question! In the context of forming a 5-digit number, the first digit cannot be zero because if it were, the number would effectively become a 4-digit number (e.g., $02347$ is just $2347$). To maintain the 5-digit requirement, we must exclude zero from the first position.

Q: Why do we use $6^4$ instead of permutations like $6 \times 5 \times 4 \times 3$?

Excellent observation! We use $6^4$ because the problem explicitly states that 'repetition is allowed.' If repetition were not allowed, we would use permutations. Since we can reuse digits, each of the four remaining slots has $6$ independent choices, leading to $6 \times 6 \times 6 \times 6 = 6^4$.

Q: Why do we stop counting at $42923$?

We stop at $42923$ because that is our target number. The rank is defined as the position of the number in the sorted list. By counting all the numbers that come *before* $42923$ and then adding $1$ (or simply counting up to it), we find its exact serial number.

Question

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are a librarian in a world where numbers are the only books on the shelves. You have a set of digits: $S = \{0, 2, 3, 4, 7, 9\}$.

Your task is to arrange all possible 5-digit numbers in ascending order and find the exact position, or 'rank,' of the number $42923$. This is a journey through the logic of combinatorics.

The first rule of our library is that a 5-digit number cannot start with zero. If it did, it would be a 4-digit number in disguise. Thus, our first digit must be chosen from $\{2, 3, 4, 7, 9\}$.

### The Sweep of Smaller Prefixes

To find the rank of $42923$, we must count every single number that comes before it. We start with the smallest possible starting digits.

**Numbers starting with $2$:**
If the first digit is $2$, the remaining four slots can be filled by any of the $6$ digits because repetition is allowed. The number of such combinations is:
$$6^4 = 1296$$
These $1296$ numbers are all smaller than our target.

**Numbers starting with $3$:**
Similarly, if the first digit is $3$, we again have $6^4 = 1296$ possibilities. Adding these to our previous count, we have accounted for:
$$1296 + 1296 = 2592 	ext{ numbers.}$$

### The Precision of the Prefix $4$

Now we reach the numbers starting with $4$. Our target $42923$ also starts with $4$, so we must be careful. We lock the first digit as $4$ and look at the second digit.

**Numbers starting with $40$:**
With the first two digits fixed as $40$, we have three slots left, giving us:
$$6^3 = 216 	ext{ numbers.}$$

**Numbers starting with $42$:**
The third digit of our target is $9$. We need to count all numbers starting with $42x$ where $x < 9$. The possible values for $x$ are $\{0, 2, 3, 4, 7\}$, which provides $5$ options. For each option, the remaining two slots can be filled in $6^2 = 36$ ways:
$$5 	imes 36 = 180 	ext{ numbers.}$$

### The Final Countdown

We have fixed the prefix $429$. The fourth digit of our target is $2$. We need the fourth digit to be less than $2$. The only option is $0$.

**Numbers starting with $4290$:**
With four slots fixed, the last slot can be any of the $6$ digits, giving us $6$ numbers.

**The final sequence:**
Finally, we look at the prefix $4292$. We list them: $42920$, $42922$, and $42923$. Our target is the $3$rd number in this final sequence.

### The Grand Total

Summing these up, we calculate the total rank:
$$1296 + 1296 + 216 + 180 + 6 + 3 = 2997$$

The elegance of this method lies in its structure. By breaking the problem into smaller, manageable buckets, we turn a daunting task into a simple, logical progression.

The rank of $42923$ is **$2997$**.

Animated Solution for Mathematics - Permutations and Combinations: Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ____.

Enter Numerical Value:

Visualized Solution

Understanding the Objective

The Dictionary Strategy

Numbers Starting with $2$

Counting Prefix $2$

Numbers Starting with $3$

Counting Prefix $3$

Numbers Starting with $40$

Counting Prefix $40$

Numbers Starting with $42 x$ ( $x < 9$ )

Counting Prefix $42 x$

Numbers Starting with $4290$

Counting Prefix $4290$

The Final Stretch

Listing the Final Numbers

Final Summation

The Sigma Insight: Linear Permutations

Analyzing the Setup

The Sweep of Smaller Prefixes

The Precision of the Prefix $4$

The Final Countdown

The Grand Total

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