We are given the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. We want to form a four-digit number using these digits. a) How many ways can we form a four-digit number? b) How many ways can we form a four-digit even number? c) How many ways can we form a four-digit odd number? d) How many ways can we form a four-digit number that is a multiple of 5? e) How many ways can we form a four-digit number with identical digits?

1. Problem Description

We are given the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and

9. We want to form a four-digit number using these digits.

a) How many ways can we form a four-digit number?

b) How many ways can we form a four-digit even number?

c) How many ways can we form a four-digit odd number?

d) How many ways can we form a four-digit number that is a multiple of 5?

e) How many ways can we form a four-digit number with identical digits?

2. Solution Steps

a) To form a four-digit number, the first digit cannot be

0. So there are 9 choices for the first digit. The other three digits can be any of the 10 digits. Therefore, the number of ways to form a four-digit number is $9 \times 10 \times 10 \times 10 = 9000$.

b) To form a four-digit even number, the last digit must be 0, 2, 4, 6, or

8. So there are 5 choices for the last digit. The first digit cannot be 0, so there are 9 choices for the first digit. The second and third digits can be any of the 10 digits. Therefore, the number of ways to form a four-digit even number is $9 \times 10 \times 10 \times 5 = 4500$.

c) To form a four-digit odd number, the last digit must be 1, 3, 5, 7, or

9. So there are 5 choices for the last digit. The first digit cannot be 0, so there are 9 choices for the first digit. The second and third digits can be any of the 10 digits. Therefore, the number of ways to form a four-digit odd number is $9 \times 10 \times 10 \times 5 = 4500$.

d) To form a four-digit number that is a multiple of 5, the last digit must be 0 or

5. So there are 2 choices for the last digit. The first digit cannot be 0, so there are 9 choices for the first digit. The second and third digits can be any of the 10 digits. Therefore, the number of ways to form a four-digit number that is a multiple of 5 is $9 \times 10 \times 10 \times 2 = 1800$.

e) To form a four-digit number with identical digits, the digit can be any digit from 1 to

9. The digit cannot be 0 because the first digit cannot be

0. Therefore, there are 9 such numbers: 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888,

9

9

9

9.

3. Final Answer

a) 9000

b) 4500

c) 4500

d) 1800

e) 9