Answered

Make the greatest and smallest four-digit numbers by using any four different digits, with the conditions given below:

1. Digit 6 is always in the thousands place.
2. Digit 4 is always in the hundreds place.
3. Digit 7 is always in the tens place.
4. Digit 1 is always in the ones place.
5. Digit 9 is always in the thousands place.
6. Digit 0 is always in the hundreds place.
7. Digit 5 is always in the tens place.
8. Digit 3 is always in the ones place.

How many numbers in all do we have with:

A. 4 digits?
B. 6 digits?



Answer :

GinnyAnswer
### Four-Digit Numbers

Let's start by constructing the greatest and the smallest four-digit numbers with the given conditions:

#### Conditions:
1. Digit 6 is always in the thousands place.
2. Digit 4 is always in the hundreds place.
3. Digit 7 is always in the tens place.
4. Digit 1 is always in the ones place.

Since all the given positions are fixed, the greatest and the smallest four-digit number would be the same.

Greatest Four-Digit Number:
- Thousands place: 6
- Hundreds place: 4
- Tens place: 7
- Ones place: 1

Therefore, the greatest four-digit number is 6471.

Smallest Four-Digit Number:
As the digits at all positions are fixed, the smallest four-digit number is also 6471.

So, the greatest and smallest four-digit numbers are:
- Greatest: 6471
- Smallest: 6471

### Six-Digit Numbers

Now, let's construct the greatest and the smallest six-digit numbers with the given conditions:

#### Conditions:
1. Digit 9 is always in the thousands place.
2. Digit 0 is always in the hundreds place.
3. Digit 5 is always in the tens place.
4. Digit 3 is always in the ones place.
5. The remaining two positions (ten-thousands and hundred-thousands) are to be filled to form the greatest and the smallest six-digit numbers.

Greatest Six-Digit Number:
To form the greatest number, we should use the largest available digits for the remaining positions, which are 8 and 7.

Thus:
- Hundred-thousands place: 9
- Ten-thousands place: 8
- Thousands place: 7
- Hundreds place: 0
- Tens place: 5
- Ones place: 3

Therefore, the greatest six-digit number is 987053.

Smallest Six-Digit Number:
To form the smallest number, we should use the smallest available digits for the remaining positions, which are 1 and 2.

Thus:
- Hundred-thousands place: 9
- Ten-thousands place: 1
- Thousands place: 0
- Hundreds place: 2
- Tens place: 5
- Ones place: 3

Therefore, the smallest six-digit number is 910253.

So, the greatest and smallest six-digit numbers are:
- Greatest: 987053
- Smallest: 910253

### Summary
To summarize, the results are:
- For four-digit numbers:
- Greatest: 6471
- Smallest: 6471
- For six-digit numbers:
- Greatest: 987053
- Smallest: 910253

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