Number Digits Reordering Analysis
The question requires us to analyze the number $8731245$. The main goal is to determine how many digits hold their original positions after the digits of the number are rearranged in ascending order from left to right.
Original Number Digits and Positions
The starting number is $8731245$. Let's identify each digit and its corresponding position:
- Position 1: The digit is $8$
- Position 2: The digit is $7$
- Position 3: The digit is $3$
- Position 4: The digit is $1$
- Position 5: The digit is $2$
- Position 6: The digit is $4$
- Position 7: The digit is $5$
Ascending Order Arrangement
Next, we take all the digits from the original number {$8$, $7$, $3$, $1$, $2$, $4$, $5$} and arrange them in ascending order (from smallest to largest).
The sequence of digits sorted in ascending order is {$1$, $2$, $3$, $4$, $5$, $7$, $8$}.
Comparing Original vs. Sorted Positions
To find out which digits remain in their original places, we compare the original number's digits with the sorted number's digits position by position. A table helps visualize this comparison:
| Original Digit | Original Position | Sorted Digit | Sorted Position | Position Unchanged? |
|---|---|---|---|---|
| $8$ | $1^{st}$ | $1$ | $1^{st}$ | No |
| $7$ | $2^{nd}$ | $2$ | $2^{nd}$ | No |
| $3$ | $3^{rd}$ | $3$ | $3^{rd}$ | Yes |
| $1$ | $4^{th}$ | $4$ | $4^{th}$ | No |
| $2$ | $5^{th}$ | $5$ | $5^{th}$ | No |
| $4$ | $6^{th}$ | $7$ | $6^{th}$ | No |
| $5$ | $7^{th}$ | $8$ | $7^{th}$ | No |
Counting Unchanged Digits
We carefully examine the table and the comparison:
- Position 1: Original digit $8$ is compared to sorted digit $1$. They are different.
- Position 2: Original digit $7$ is compared to sorted digit $2$. They are different.
- Position 3: Original digit $3$ is compared to sorted digit $3$. They are the same.
- Position 4: Original digit $1$ is compared to sorted digit $4$. They are different.
- Position 5: Original digit $2$ is compared to sorted digit $5$. They are different.
- Position 6: Original digit $4$ is compared to sorted digit $7$. They are different.
- Position 7: Original digit $5$ is compared to sorted digit $8$. They are different.
From this step-by-step comparison, we find that only the digit $3$ remains in its original position, which is the 3rd position.
Conclusion on Unchanged Positions
Therefore, the total count of digits whose positions do not change after arranging the number's digits in ascending order is one.
Important Questions from Ordering and Ranking
- Ariel ranked \(6^{th}\) from the top and \(18^{th}\) from the bottom in her class. How many students are there in the class?
- Varun ranked 34th from the top and 11th from the bottom in his class. How many students are there in his class?
- Piyush ranked 35th from the top and 18th from the bottom in his class. How many students are there in his class?
- Each of the digits in the number 81674293 is arranged in ascending order from left to right. Which of the following digits will be fourth from the left?
- Mr. Into ranked 96th from the top and 31st from the bottom in his class. How many students are there in his class?
