Multiple People, Overlaps & Interchanges

🎯 Learning Objective

Count people between positions, handle overlapping ranks from opposite ends, and solve interchange problems.

💡 Concept

• Convert everyone to the SAME reference end first, then read their positions off one line
• People strictly between two positions = |difference of positions| − 1
• Overlapping ranks (from opposite ends, with L + R > Total + 1) mean the two counts have crossed — convert one rank to the other end first
• Interchange: when two people swap seats, each takes the other's EXACT position
• After an interchange, apply Total = new-left + new-right − 1 on the shared seat

🧮 Key Formulas

People between two positions = |difference| − 1

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Convert ends: rank from left = Total − rank from right + 1

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After interchange: Total = new-left + new-right − 1

✏️ Easy Example

Q. In a queue, Arun is 10th from the front and Bina is 20th from the front. How many people stand between them?

1. Both are counted from the same end (front)
2. People between = 20 − 10 − 1

Answer: 9 people

🇮🇳 Real-Life Example

Two friends checking their exam roll positions from the noticeboard — 'tu 14th, main 11th, beech mein do log' — that beech-wala count is exactly this lesson.

📝 Exam-Level Example

Q. In a row of 25 children, Anil is 14th from the left and Bimal is 15th from the right. How many children sit between them?

1. 14 + 15 = 29 > 25 → the two counts overlap (they have crossed)
2. Convert Bimal to the left end: 25 − 15 + 1 = 11th from the left
3. Anil is 14th, Bimal is 11th from the left → between = 14 − 11 − 1

Answer: 2 children (the 12th and 13th)

📝 Exam-Level Example

Q. In a row, Priya is 7th from the left and Rohan is 9th from the right. They interchange their positions, and now Priya becomes 11th from the left. How many people are in the row?

1. After the swap, Priya sits in Rohan's old seat → Priya is now 9th from the right
2. Priya is now also 11th from the left (given)
3. That single seat gives Total = 11 + 9 − 1

Answer: 19 people

🪄 Memory Trick

For interchange questions, remember only this: the person who moves takes the OTHER person's old position. Combine that new position with the given new rank and apply Total = L + R − 1.

⚠️ Common Mistakes

• ❌ Using |difference| − 1 on ranks taken from opposite ends without converting first
• ❌ Assuming no overlap when L + R already exceeds the total
• ❌ In interchange, forgetting that the mover inherits the other's exact seat

🏆 Exam Tips

• ✅ Always convert to a single reference end before counting people in between
• ✅ If L + R > Total + 1, the counts overlap — expect a small between-number

📌 Summary

• Convert to one end, then between = |difference| − 1
• Overlap (L + R > Total + 1) → convert one rank first
• Interchange: mover takes the other's exact seat
• Re-apply Total = L + R − 1 on that seat