Try These

1. Use the given digits without repetition to make the greatest and smallest 4-digit numbers:
a. 2, 8, 7, 4
b. 9, 7, 4, 1
c. 4, 7, 5, 0
d. 1, 7, 6, 2
e. 5, 4, 0, 3
(Hint: 0754 is a 3-digit number.)

2. Now, make the greatest and smallest 4-digit numbers by using any one digit twice:
a. 3, 8, 7
b. 9, 0, 5
c. 0, 4, 9
d. 8, 5, 1
(Hint: Think in each case which digit you will use twice.)

Certainly! Let's go through each part of the question step-by-step and determine the answers.

### Part 1: Use the given digits without repetition and make the greatest and smallest 4-digit numbers.

#### (a) Digits: 2, 8, 7, 4
- Greatest 4-digit number: Arrange the digits in descending order: 8, 7, 4, 2. Thus, the greatest number is 8742.
- Smallest 4-digit number: Arrange the digits in ascending order: 2, 4, 7, 8. Thus, the smallest number is 2478.

#### (b) Digits: 9, 7, 4, 1
- Greatest 4-digit number: Arrange the digits in descending order: 9, 7, 4, 1. Thus, the greatest number is 9741.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 4, 7, 9. Thus, the smallest number is 1479.

#### (c) Digits: 4, 7, 5, 0
- Greatest 4-digit number: Arrange the digits in descending order: 7, 5, 4, 0. Thus, the greatest number is 7540.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 4, 5, 7. Since 0457 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 4, 5, 0, 7. Thus, the smallest number is 4570.

#### (d) Digits: 1, 7, 6, 2
- Greatest 4-digit number: Arrange the digits in descending order: 7, 6, 2, 1. Thus, the greatest number is 7621.
- Smallest 4-digit number: Arrange the digits in ascending order: 1, 2, 6, 7. Thus, the smallest number is 1267.

#### (e) Digits: 5, 4, 0, 3
- Greatest 4-digit number: Arrange the digits in descending order: 5, 4, 3, 0. Thus, the greatest number is 5430.
- Smallest 4-digit number: Arrange the digits in ascending order: 0, 3, 4, 5. Since 0345 is not a 4-digit number, we need the smallest arrangement of these digits without leading zero: 3, 4, 5, 0. Thus, the smallest number is 3450.

### Part 2: Make the greatest and smallest 4-digit numbers by using any one digit twice.

#### (a) Digits: 3, 8, 7
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 7, 3. Thus, the greatest number is 8873.
- Smallest 4-digit number: Duplicate the smallest digit (3) and arrange in ascending order: 3, 3, 7, 8. Thus, the smallest number is 3378.

#### (b) Digits: 9, 0, 5
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 5, 0. Thus, the greatest number is 9950.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 5, 9. Since 0059 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 5, 9, 0. Thus, the smallest number is 0590. To fix the leading zero issue, rearrange to 590.

#### (c) Digits: 0, 4, 9
- Greatest 4-digit number: Duplicate the largest digit (9) and arrange in descending order: 9, 9, 4, 0. Thus, the greatest number is 9940.
- Smallest 4-digit number: Duplicate the smallest digit (0) and arrange in ascending order: 0, 0, 4, 9. Since 0049 is not a 4-digit number, we need the smallest arrangement without leading zero: 0, 4, 9, 0. Thus, the smallest number is 490.

#### (d) Digits: 8, 5, 1
- Greatest 4-digit number: Duplicate the largest digit (8) and arrange in descending order: 8, 8, 5, 1. Thus, the greatest number is 8851.
- Smallest 4-digit number: Duplicate the smallest digit (1) and arrange in ascending order: 1, 1, 5, 8. Thus, the smallest number is 1158.

These solutions provide the greatest and smallest 4-digit numbers for each set of digits both without repetition and by using one digit twice.