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Answer :
To find out how many numbers lie between 1000 and 10000 that can be formed using the digits [tex]\(1, 2, 3, 4, 5, 6\)[/tex] with repetition allowed, let's break it down step by step.
1. Determine the Number of Digits:
- Numbers between 1000 and 10000 are 4-digit numbers.
2. Choose Digits for Each Place:
- Since repetition of digits is allowed, each digit (thousands, hundreds, tens, and units) can be any of the six given digits (1, 2, 3, 4, 5, 6).
3. Calculate Possible Combinations:
- For the thousands place, there are 6 options (1, 2, 3, 4, 5, 6).
- For the hundreds place, there are also 6 options.
- For the tens place, there are again 6 options.
- For the units place, there are 6 options.
4. Total Number of Combinations:
- To find the total number of 4-digit numbers that can be formed, you multiply the number of choices for each digit place.
[tex]\[
\text{Total Combinations} = 6 \times 6 \times 6 \times 6 = 6^4
\][/tex]
5. Result:
- The total number of different 4-digit numbers you can form is [tex]\(6^4 = 1296\)[/tex].
So, there are 1296 numbers that can be formed as described.
1. Determine the Number of Digits:
- Numbers between 1000 and 10000 are 4-digit numbers.
2. Choose Digits for Each Place:
- Since repetition of digits is allowed, each digit (thousands, hundreds, tens, and units) can be any of the six given digits (1, 2, 3, 4, 5, 6).
3. Calculate Possible Combinations:
- For the thousands place, there are 6 options (1, 2, 3, 4, 5, 6).
- For the hundreds place, there are also 6 options.
- For the tens place, there are again 6 options.
- For the units place, there are 6 options.
4. Total Number of Combinations:
- To find the total number of 4-digit numbers that can be formed, you multiply the number of choices for each digit place.
[tex]\[
\text{Total Combinations} = 6 \times 6 \times 6 \times 6 = 6^4
\][/tex]
5. Result:
- The total number of different 4-digit numbers you can form is [tex]\(6^4 = 1296\)[/tex].
So, there are 1296 numbers that can be formed as described.
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