1.      **Question. -**  If you have 150 pebbles and need to make packets out of it in such a way that if any number of pebbles up to 150 is required you can hand it over without opening the packets, what is the minimum number of packets you need?

**Options. -**

a)      10

b)      11

c)      12

d)      None of the given options

To solve this problem, we need to find the minimum number of packets, each containing a different number of pebbles, such that any number of pebbles from 1 to 150 can be formed using a combination of these packets.

The key idea here is to use powers of 2. If we create packets with quantities as powers of 2, we can cover all numbers up to the sum of these quantities.

### Step-by-step explanation:

1. **Powers of 2 sequence**:
   - 2^0 = 1
   - 2^1 = 2
   - 2^2 = 4
   - 2^3 = 8
   - 2^4 = 16
   - 2^5 = 32
   - 2^6 = 64
   - 2^7 = 128

The sum of these values from 2^0 to 2^7 is:
\[ 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 \]

Since 255 is greater than 150, we can cover all the numbers from 1 to 150 using these powers of 2.

### Conclusion:
We need 8 packets to achieve this. Since none of the given options match this number, the correct answer is:

**d) None of the given options**

Would you like more details or have any questions about this solution?

### Related Questions:
1. How can binary representation be used to form different numbers with a minimum number of packets?
2. What would be the minimum number of packets required if we had 1000 pebbles instead?
3. Can the same approach be used if the numbers required are in a different range (e.g., 1-200)?
4. What other sequences, besides powers of 2, could be used to solve this problem?
5. How can this method be applied in computer science for data representation?

**Tip:** Using powers of 2 allows for efficient representation of any integer within a given range, similar to binary coding in computer systems.

If you have 150 pebbles and need to make packets out of it in such a way that if any number of pebbles up to 150 is required you can hand it over without opening the packets, what is the minimum number of packets you need?

Discover how to solve this problem using powers of two to create packets that can form any quantity from 1 to 150 pebbles. Learn about binary decomposition and combinatorial principles to minimize packet count. Suitable for Grades 8-10.

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