Understanding Digit Position Changes in Number Arrangement

The question asks us to identify how many digits in the number 4512689 maintain their original positions after the digits are rearranged in ascending order.

Original Number Analysis

The original number given is 4512689. Let's list the digits and their positions:

• Position 1: Digit 4
• Position 2: Digit 5
• Position 3: Digit 1
• Position 4: Digit 2
• Position 5: Digit 6
• Position 6: Digit 8
• Position 7: Digit 9

Arranging Digits in Ascending Order

Next, we need to take the digits from the original number (4, 5, 1, 2, 6, 8, 9) and arrange them in ascending (smallest to largest) order.

The digits are: 1, 2, 4, 5, 6, 8, 9.

Arranging these in ascending order gives us the new sequence: 1245689.

Comparing Original and Ascending Order Positions

Now, we compare the digits in each position of the original number with the digits in the corresponding position of the newly arranged number (1245689).

Original Number: $4$ $5$ $1$ $2$ $6$ $8$ $9$

Ascending Order: $1$ $2$ $4$ $5$ $6$ $8$ $9$

Let's check each position:

• Position 1: Original digit is $4$, Ascending order digit is $1$. Positions are different.
• Position 2: Original digit is $5$, Ascending order digit is $2$. Positions are different.
• Position 3: Original digit is $1$, Ascending order digit is $4$. Positions are different.
• Position 4: Original digit is $2$, Ascending order digit is $5$. Positions are different.
• Position 5: Original digit is $6$, Ascending order digit is $6$. Positions are the same.
• Position 6: Original digit is $8$, Ascending order digit is $8$. Positions are the same.
• Position 7: Original digit is $9$, Ascending order digit is $9$. Positions are the same.

Determining the Number of Unchanged Positions

By comparing the digits in their respective positions, we found that the digits in the 5th, 6th, and 7th positions remain unchanged.

These unchanged digits are $6$, $8$, and $9$.

Therefore, the count of digits whose positions remain unchanged is 3.