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Sagot :
Alright, let's go through this problem step-by-step and fill in the table with the correct values for each set of digits.
### Digits: 4, 0
- Smallest 2-digit number: Arrange the digits in ascending order: 0 and 4 make 40, but since '04' is not valid, the smallest possible number using both digits and respecting numerical order is 4.
- Greatest 2-digit number: Arrange the digits in descending order: 4 and 0 make 40.
- Smallest 3-digit number: Not applicable (N/A) since we only have 2 digits.
- Greatest 3-digit number: Not applicable (N/A) since we only have 2 digits.
### Digits: 5, 1
- Smallest 2-digit number: Arrange the digits in ascending order: 1 and 5 make 15.
- Greatest 2-digit number: Arrange the digits in descending order: 5 and 1 make 51.
- Smallest 3-digit number: Not applicable (N/A) since we only have 2 digits.
- Greatest 3-digit number: Not applicable (N/A) since we only have 2 digits.
### Digits: 3, 6, 9
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 3 and 6 make 36.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 6 and 3 make 63.
- Smallest 3-digit number: Arrange all digits in ascending order: 3, 6, and 9 make 369.
- Greatest 3-digit number: Arrange all digits in descending order: 9, 6, and 3 make 963.
### Digits: 2, 0, 7
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 0 and 2 make 20, but '02' is not valid, so next digits 2 and 7 make 27.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 7 and 2 make 72, but 720 is greater.
- Smallest 3-digit number: Arrange all digits in ascending order: 0, 2, and 7 make 207.
- Greatest 3-digit number: Arrange all digits in descending order: 7, 2, and 0 make 720.
### Digits: 9, 8, 1
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 1 and 8 make 18, but 89 is smaller.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 9 and 8 make 98.
- Smallest 3-digit number: Arrange all digits in ascending order: 1, 8, and 9 make 189.
- Greatest 3-digit number: Arrange all digits in descending order: 9, 8, and 1 make 981.
Now, let's fill in the table with these results:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Digits} & \text{Smallest 2/3-digit number} & \text{Greatest 2/3-digit number} \\ \hline 4,0 & 4 / N/A & 40 / N/A \\ \hline 5,1 & 15 / N/A & 51 / N/A \\ \hline 3,6,9 & 36 / 369 & 63 / 963 \\ \hline 2,0,7 & 2 / 20 & 27 / 720 \\ \hline 9,8,1 & 89 / 98 & 189 / 981 \\ \hline \end{tabular} \][/tex]
Here is the filled table with all the values properly calculated and entered.
### Digits: 4, 0
- Smallest 2-digit number: Arrange the digits in ascending order: 0 and 4 make 40, but since '04' is not valid, the smallest possible number using both digits and respecting numerical order is 4.
- Greatest 2-digit number: Arrange the digits in descending order: 4 and 0 make 40.
- Smallest 3-digit number: Not applicable (N/A) since we only have 2 digits.
- Greatest 3-digit number: Not applicable (N/A) since we only have 2 digits.
### Digits: 5, 1
- Smallest 2-digit number: Arrange the digits in ascending order: 1 and 5 make 15.
- Greatest 2-digit number: Arrange the digits in descending order: 5 and 1 make 51.
- Smallest 3-digit number: Not applicable (N/A) since we only have 2 digits.
- Greatest 3-digit number: Not applicable (N/A) since we only have 2 digits.
### Digits: 3, 6, 9
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 3 and 6 make 36.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 6 and 3 make 63.
- Smallest 3-digit number: Arrange all digits in ascending order: 3, 6, and 9 make 369.
- Greatest 3-digit number: Arrange all digits in descending order: 9, 6, and 3 make 963.
### Digits: 2, 0, 7
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 0 and 2 make 20, but '02' is not valid, so next digits 2 and 7 make 27.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 7 and 2 make 72, but 720 is greater.
- Smallest 3-digit number: Arrange all digits in ascending order: 0, 2, and 7 make 207.
- Greatest 3-digit number: Arrange all digits in descending order: 7, 2, and 0 make 720.
### Digits: 9, 8, 1
- Smallest 2-digit number: Consider only two digits and arrange them in ascending order: 1 and 8 make 18, but 89 is smaller.
- Greatest 2-digit number: Consider only two digits and arrange them in descending order: 9 and 8 make 98.
- Smallest 3-digit number: Arrange all digits in ascending order: 1, 8, and 9 make 189.
- Greatest 3-digit number: Arrange all digits in descending order: 9, 8, and 1 make 981.
Now, let's fill in the table with these results:
[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Digits} & \text{Smallest 2/3-digit number} & \text{Greatest 2/3-digit number} \\ \hline 4,0 & 4 / N/A & 40 / N/A \\ \hline 5,1 & 15 / N/A & 51 / N/A \\ \hline 3,6,9 & 36 / 369 & 63 / 963 \\ \hline 2,0,7 & 2 / 20 & 27 / 720 \\ \hline 9,8,1 & 89 / 98 & 189 / 981 \\ \hline \end{tabular} \][/tex]
Here is the filled table with all the values properly calculated and entered.
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