The positions of the first and sixth digits in the number 5109238674 are interchanged, the positions of the second and seventh digits are interchanged, the positions of the third and eighth digits are interchanged, and so on. After this rearrangement, which digit will be the third digit from the right end of the new number?

Introduction / Context:
This question involves a specific pattern of interchanging digits in a long number and then identifying the digit at a particular position after the rearrangement. It tests attention to detail and the ability to follow a structured swap of positions correctly across the entire number.

Given Data / Assumptions:

• Original number: 5109238674.
• Digits are numbered from left to right as positions 1 to 10.
• Position 1 is swapped with position 6, position 2 with position 7, position 3 with position 8, position 4 with position 9, and position 5 with position 10.
• We must determine the third digit from the right end after all these swaps are done.

Concept / Approach:
The procedure is to write the positions and digits clearly, perform each indicated swap exactly once, and then read off the resulting number. Once the new number is known, counting from the right to find the third digit is straightforward. Careful bookkeeping of each interchange is important to avoid confusion or double swapping the same pair.

Step-by-Step Solution:
Step 1: Write the digits with their positions from left to right: 1:5, 2:1, 3:0, 4:9, 5:2, 6:3, 7:8, 8:6, 9:7, 10:4. Step 2: Swap positions 1 and 6: digits 5 and 3 are interchanged. Now positions are 1:3, 2:1, 3:0, 4:9, 5:2, 6:5, 7:8, 8:6, 9:7, 10:4. Step 3: Swap positions 2 and 7: digits 1 and 8 are interchanged. New positions: 1:3, 2:8, 3:0, 4:9, 5:2, 6:5, 7:1, 8:6, 9:7, 10:4. Step 4: Swap positions 3 and 8: digits 0 and 6 are interchanged. New positions: 1:3, 2:8, 3:6, 4:9, 5:2, 6:5, 7:1, 8:0, 9:7, 10:4. Step 5: Swap positions 4 and 9: digits 9 and 7 are interchanged. New positions: 1:3, 2:8, 3:6, 4:7, 5:2, 6:5, 7:1, 8:0, 9:9, 10:4. Step 6: Swap positions 5 and 10: digits 2 and 4 are interchanged. Final arrangement: 1:4, 2:8, 3:6, 4:7, 5:4, 6:5, 7:1, 8:0, 9:9, 10:2. Step 7: The new number is 4867451092. Step 8: Count from the right: from rightmost, 1st digit is 2, 2nd digit is 9, 3rd digit is 0. Step 9: Therefore, the third digit from the right end is 0.

Verification / Alternative check:
You can verify by writing the final number again and labeling the positions from the right. From the right, positions are 1:2, 2:9, 3:0, 4:1, 5:5, 6:4, 7:7, 8:6, 9:8, 10:4. The third position from the right is indeed 0, confirming that no swap was misapplied.

Why Other Options Are Wrong:
9 and 6 appear close to the right end but are the second and fourth digits from the right, not the third. 3 does not appear near the relevant position in the final arrangement.

Common Pitfalls:
Mistakes usually occur when a pair is swapped twice or when positions are misnumbered. Another error is to forget that the question asks for the third digit from the right, not from the left. Drawing a simple positional table and carefully marking each swap helps avoid these common errors.

Final Answer:
The third digit from the right end after the rearrangement is 0.