Math Problem Statement
Solution
Let's solve the problem based on the details provided in the image:
Method A:
- First digit: It must be an odd number between 0 and 10. The odd numbers in this range are: 1, 3, 5, 7, 9. This gives 5 possibilities.
- Last two digits: Both must be multiples of 10. In the range from 0 to 9, the only multiple of 10 is 0. This gives only 1 possibility for each of the last two digits.
Thus, the total number of codes for Method A is:
Method B:
- First digit: It must be an odd number (but not 0). The odd numbers are: 1, 3, 5, 7, 9. This gives 5 possibilities.
- Second digit: It must be an even number (but not 0). The even numbers are: 2, 4, 6, 8. This gives 4 possibilities.
- Third digit: It must be an odd number (but not 0). The odd numbers are again: 1, 3, 5, 7, 9. This gives 5 possibilities.
Thus, the total number of codes for Method B is:
Difference:
To find how many more codes can be made with Method B than with Method A, subtract the number of Method A codes from the number of Method B codes:
Thus, Method B can make 95 more codes than Method A.
The answer is:
Would you like any further clarifications or detailed steps?
Here are 5 relative questions to explore further:
- How would the solution change if repetition of digits was not allowed?
- What if Method A allowed any digit for the last two positions instead of just multiples of 10?
- How many codes can be made if zero was allowed for the second digit in Method B?
- How does the solution change if we use 4-digit codes instead of 3-digit codes?
- Can you create your own method for generating 3-digit codes and calculate how many possible combinations exist?
Tip: Always break down problems like this into simple, logical steps, and list all possibilities to avoid missing any cases!
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Multiplication Principle of Counting
Formulas
Total Codes for Method A = (Possible First Digits) * (Possible Second Digits) * (Possible Third Digits)
Total Codes for Method B = (Possible First Digits) * (Possible Second Digits) * (Possible Third Digits)
Theorems
Multiplication Rule of Counting
Suitable Grade Level
Grades 6-8
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