Understanding the Question: Finding the Largest 5-Digit Number

The question asks for the largest possible number that has exactly five digits and is formed using only three different digits. A 5-digit number is any integer from 10000 to 99999.

Identifying the Constraint: Three Different Digits

The key constraint is that we can only use three unique digits out of the possible ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). To create the Largest 5-Digit Number, we should choose the largest possible digits available and place them in the highest value positions.

Selecting the Best Digits for the Largest Number

To make a number as large as possible, we should use the largest digits. The largest digits are 9, 8, 7, 6, and so on. Since we are limited to using only three different digits, we should pick the three largest digits available. These are 9, 8, and 7.

Constructing the Largest 5-Digit Number using 9, 8, and 7

A 5-digit number has positions for ten thousands, thousands, hundreds, tens, and units. To make the number largest, the digit in the ten-thousands place must be the largest possible. We should use the largest digit from our chosen set {9, 8, 7}, which is 9.

So, the number starts with 9xxxx.

Next, the thousands place should also be as large as possible. Using 9 again makes the number larger: 99xxx.

Similarly, the hundreds place should be 9 to maximize the value: 999xx.

We now have the number 999xx. We have used the digit 9 three times. We still need to fill the tens and units places, and we must ensure the final number uses exactly three different digits. Our chosen digits are 9, 8, and 7. We have used 9. We must now introduce 8 and 7 into the remaining places.

To make the number largest, the next largest digit from our set {8, 7} should go into the tens place. This is 8. So the number becomes 9998x.

Finally, the remaining digit from our set {7} must go into the units place. This is 7. So the number becomes 99987.

Verifying the Result

Let's check if 99987 meets the criteria:

• Is it a 5-digit number? Yes, it is 99987.
• Does it have exactly three different digits? Yes, the digits used are 9, 8, and 7. There are only three unique digits.
• Is it the largest possible number meeting these conditions? By placing the largest available digits (9, 8, 7) in the highest possible value positions ( ưu tiên 9 ở hàng vạn, nghìn, trăm, sau đó 8 ở hàng chục, rồi 7 ở hàng đơn vị), we ensure the resulting number is the Largest 5-Digit Number possible under the constraint of using only three different digits. Any other arrangement using 9, 8, and 7 would result in a smaller number (e.g., 99897, 98799, 98978 are all smaller than 99987). Using smaller digits instead of 9, 8, and 7 would also result in a smaller number.

Therefore, 99987 is the Largest 5-Digit Number having three different digits.