What is the number of $500$ rupee notes required to serve $50$ customers with $500$ rupee notes as their preferences and another $50$ customers with $100$ rupee notes as their preferences, if the total number of notes to be dispensed is the smallest possible?

For Rs. 500 Preference Customers

From previous analysis, valid combinations are:

Rs. $100$	Rs. $200$	Rs. $500$	Total Notes
$4$	$3$	$8$	$15$
$2$	$4$	$8$	$14$
$0$	$5$	$8$	$13$
$5$	$0$	$9$	$14$
$3$	$1$	$9$	$13$
$1$	$2$	$9$	$12$
$0$	$0$	$10$	$10$

Minimum notes per customer: $10$ notes ($0 \times 100 + 0 \times 200 + 10 \times 500$)

---

For Rs. 100 Preference Customers

Need $x > y+z$ where $100x + 200y + 500z = 5000$

Valid combinations:

Rs. $100$	Rs. $200$	Rs. $500$	Check: $x > y+z$	Total Notes
$50$	$0$	$0$	$50 > 0+0$ ✓	$50$
$45$	$0$	$1$	$45 > 0+1$ ✓	$46$
$40$	$5$	$0$	$40 > 5+0$ ✓	$45$
$35$	$5$	$1$	$35 > 5+1$ ✓	$41$
$30$	$10$	$0$	$30 > 10+0$ ✓	$40$
$30$	$0$	$4$	$30 > 0+4$ ✓	$34$
$25$	$10$	$1$	$25 > 10+1$ ✓	$36$
$25$	$0$	$5$	$25 > 0+5$ ✓	$30$
$20$	$15$	$0$	$20 > 15+0$ ✓	$35$
$20$	$5$	$4$	$20 > 5+4$ ✓	$29$
$20$	$0$	$6$	$20 > 0+6$ ✓	$26$
$15$	$10$	$3$	$15 > 10+3$ ✓	$28$
$15$	$0$	$7$	$15 > 0+7$ ✓	$22$
$10$	$15$	$2$	$10 > 15+2$ ✗	Invalid
$10$	$5$	$6$	$10 > 5+6$ ✗	Invalid
$10$	$0$	$8$	$10 > 0+8$ ✓	$18$
$5$	$0$	$9$	$5 > 0+9$ ✗	Invalid

Minimum notes per customer: $18$ notes ($10 \times 100 + 0 \times 200 + 8 \times 500$)

---

Optimal Strategy

For $50$ customers with Rs. $500$ preference:

• Use: $0 \times 100 + 0 \times 200 + 10 \times 500 = 10$ notes each
• Rs. $500$ notes per customer: $10$
• Total Rs. $500$ notes: $50 \times 10 = 500$

For $50$ customers with Rs. $100$ preference:

• Use: $10 \times 100 + 0 \times 200 + 8 \times 500 = 18$ notes each
• Rs. $500$ notes per customer: $8$
• Total Rs. $500$ notes: $50 \times 8 = 400$

---

Final Answer

Total Rs. $500$ notes required $= 500 + 400 = 900$