To find the minimum number of bags required to form any sum up to a certain total, we use the principle of powers of 2 (binary representation).
Step 1: Understand the Principle
To represent any number from 1 to N using the minimum number of components, the components should ideally be in the form of 20,21,22,…,2n−1. The sum of a geometric series of powers of 2 is:
Where n is the number of components (bags).
Step 2: Apply the condition to Rs. 1074
We need to find the smallest value of n such that the sum 2n−1 is greater than or equal to 1074.
Step 3: Calculate the power of 2
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If n=9, then 29=512 (This is less than 1075).
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If n=10, then 210=1024 (This is still less than 1075).
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If n=11, then 211=2048 (This is greater than 1075).
Step 4: Conclusion
Since 210 (1024) is not enough to reach 1074, we need at least 11 bags. With 11 bags, we can represent any amount up to 2047, which covers the required range of 1 to 1074.
The bags would contain amounts like 1,2,4,8,16,32,64,128,256,512 and the remaining amount to reach the total.
The minimum number of bags required is 11.
Correct Option: (d)
