The problem states that we have the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. We want to form four-digit numbers with distinct digits. a) How many four-digit numbers can be formed? b) How many four-digit even numbers can be formed? c) How many four-digit odd numbers can be formed? d) How many four-digit numbers divisible by 5 can be formed?

1. Problem Description

The problem states that we have the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. We want to form four-digit numbers with distinct digits.

a) How many four-digit numbers can be formed?

b) How many four-digit even numbers can be formed?

c) How many four-digit odd numbers can be formed?

d) How many four-digit numbers divisible by 5 can be formed?

2. Solution Steps

a) Total number of four-digit numbers with distinct digits:

For the first digit, we have 9 choices (1-9, since it can't be 0).

For the second digit, we have 9 choices (0 and the remaining 8 digits).

For the third digit, we have 8 choices.

For the fourth digit, we have 7 choices.

Therefore, the total number of four-digit numbers is $9 \times 9 \times 8 \times 7 = 4536$.

b) Total number of four-digit even numbers with distinct digits:

Case 1: The last digit is

0. We have 9 choices for the first digit, 8 for the second, and 7 for the third. So, $9 \times 8 \times 7 = 504$ such numbers.

Case 2: The last digit is 2, 4, 6, or

8. We have 4 choices for the last digit.

The first digit can't be 0 or the digit chosen for the last digit, so we have 8 choices.

For the second digit, we can have 0, but not the digits used for the first and last digits, so we have 8 choices.

For the third digit, we have 7 choices.

So, $4 \times 8 \times 8 \times 7 = 1792$.

The total number of four-digit even numbers is $504 + 1792 = 2296$.

c) Total number of four-digit odd numbers with distinct digits:

The total number of four-digit numbers is

4

5

3

6. The number of even four-digit numbers is

2

2

9

6. So the number of odd four-digit numbers is $4536 - 2296 = 2240$.

d) Total number of four-digit numbers divisible by 5 with distinct digits:

Case 1: The last digit is

0. We have 9 choices for the first digit, 8 for the second, and 7 for the third. So, $9 \times 8 \times 7 = 504$ such numbers.

Case 2: The last digit is

5. We have 8 choices for the first digit (can't be 0 or 5).

We have 8 choices for the second digit (can be 0, but not 5 or the first digit).

We have 7 choices for the third digit.

So, $8 \times 8 \times 7 = 448$.

The total number of four-digit numbers divisible by 5 is $504 + 448 = 952$.

3. Final Answer

a) 4536

b) 2296

c) 2240

d) 952