Permutations and Combinations

To find the difference between the number of odd and even numbers formed by the digits 3,5,6, 7 , and 8 , where the numbers are greater than 6000 and less than 10000 , we consider the following process:
Odd Numbers:
Set I: When the first digit (thousands place) is 6,7 , or 8 , and the last digit (units place) is 3 .
Choices:
3 choices for the first digit ( 6,7,8 ).
3 choices for the second digit.
2 choices for the third digit.
1 choice for the last digit (3).
Combinations:
3×3×2×1=18
Set II: Similar to Set I, but the last digit (units place) is 5.
Choices:
3 choices for the first digit (6,7,8).
3 choices for the second digit.
2 choices for the third digit.
To find the difference between the number of odd and even numbers formed by the digits 3,5,6, 7 , and 8 , where the numbers are greater than 6000 and less than 10000 , we consider the following process:
Odd Numbers:
Set I: When the first digit (thousands place) is 6,7 , or 8 , and the last digit (units place) is 3 .
Choices:
3 choices for the first digit ( 6,7,8 ).
3 choices for the second digit.
2 choices for the third digit.
1 choice for the last digit (3).
Combinations:
3×3×2×1=18
Set II: Similar to Set I, but the last digit (units place) is 5.
Choices:
3 choices for the first digit (6,7,8).
3 choices for the second digit.
2 choices for the third digit.
1 choice for the last digit (8).
Combinations:
2×3×2×1=12
Total Even Numbers:
12+12=24
Calculating the Difference:
The required difference between the number of odd and even numbers is:
48−24=24
This result can be expressed as ‌4P3.