Explanation

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 \times 3 \times 2 \times 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 (5).

Combinations:

$ 3 \times 3 \times 2 \times 1 = 18 $

Set III: The first digit (thousands place) is 6 or 8, and the last digit is 7.

Choices:

2 choices for the first digit (6, 8).

3 choices for the second digit.

2 choices for the third digit.

1 choice for the last digit (7).

Combinations:

$ 2 \times 3 \times 2 \times 1 = 12 $

Total Odd Numbers:

$ 18 + 18 + 12 = 48 $

Even Numbers:

Set I: The first digit (thousands place) is 7 or 8, and the last digit (units place) is 6.

Choices:

2 choices for the first digit (7, 8).

3 choices for the second digit.

2 choices for the third digit.

1 choice for the last digit (6).

Combinations:

$ 2 \times 3 \times 2 \times 1 = 12 $

Set II: The first digit (thousands place) is 6 or 7, and the last digit (units place) is 8.

Choices:

2 choices for the first digit (6, 7).

3 choices for the second digit.

2 choices for the third digit.

1 choice for the last digit (8).

Combinations:

$ 2 \times 3 \times 2 \times 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 $ {}^4 P_3 $.

Permutations And Combinations

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