Maximum Rs. 500 Preference Customers

Key Insight

The question asks for maximum total customers, not maximum customers preferring Rs. $500$ notes.

Strategy: Minimize Rs. 500 notes per customer

To serve maximum customers, find the minimum Rs. $500$ notes needed per customer while satisfying:

1. At most $20$ notes total
2. Preferred denomination dominates

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Case Analysis

Rs. 100 Preference ($x > y+z$)

Need minimum Rs. $500$ notes while keeping total $\leq 20$.

• Testing $z=0$: $100x+200y=5000 \;\Rightarrow\; x+2y=50$

Minimum satisfying $x>y$: $(x=34,y=8)$ → $42$ notes $>20$ ✗

• Testing $z=2$: $100x+200y=4000 \;\Rightarrow\; x+2y=40$

Try $(x=20,y=10)$: $20>10+2=12$ ✓, total $=32$ notes $>20$ ✗

• Testing $z=4$: $100x+200y=3000 \;\Rightarrow\; x+2y=30$

Try $(x=16,y=7)$: $16>7+4=11$ ✓, total $=27$ notes $>20$ ✗

• Testing $z=5$: $100x+200y=2500 \;\Rightarrow\; x+2y=25$

Try $(x=13,y=6)$: $13>6+5=11$ ✓, total $=24$ notes $>20$ ✗

• Testing $z=6$: $100x+200y=2000 \;\Rightarrow\; x+2y=20$

Try $(x=12,y=4)$: $12>4+6=10$ ✓, total $=22$ notes $>20$ ✗

• Testing $z=7$: $100x+200y=1500 \;\Rightarrow\; x+2y=15$

Try $(x=11,y=2)$: $11>2+7=9$ ✓, total $=20$ notes ✓

Rs. 100 preference minimum: $7$ Rs. $500$ notes

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Rs. 200 Preference ($y > x+z$)

• Testing $z=0$: $x+2y=50$, need $y>x$.

Minimum $(x=0,y=25)$ → $25$ notes $>20$ ✗

• Testing $z=2$: $x+2y=45$, need $y>x+2$.

Try $(x=1,y=22)$: $22>1+2=3$ ✓, total $=25$ notes $>20$ ✗

• Testing $z=4$: $x+2y=30$, need $y>x+4$.

Try $(x=0,y=15)$: $15>0+4=4$ ✓, total $=19$ notes ✓

Try $(x=2,y=14)$: $14>2+4=6$ ✓, total $=20$ notes ✓

Rs. 200 preference minimum: $4$ Rs. $500$ notes

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Rs. 500 Preference

From earlier analysis: minimum $8$ Rs. $500$ notes.

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Optimization

Minimum Rs. $500$ notes per customer by preference:

• Rs. $500$ preference: $8$ notes
• Rs. $200$ preference: $4$ notes
• Rs. $100$ preference: $7$ notes

To maximize total customers:

Use all customers with Rs. $200$ preference (each $4$ Rs. $500$ notes).

Maximum customers $= \left\lfloor \dfrac{50}{4} \right\rfloor = 12$.

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Maximum customers the ATM can serve: $12$

All $12$ customers would prefer Rs. $200$ notes, each using $4$ Rs. $500$ notes.