Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Question

Homi · Accepted Answer

To make the number largest, we pick the biggest digit for each place.

Step 1 — Finding the first digit

We need to choose the first digit of our new number. This digit should be the largest possible. We must keep 10 digits in total. We need 9 more digits after the first. The original number has 20 digits. So, the first digit must be chosen from the first $20 - 9$ digits. $20 - 9 = 11$ So, the first digit must be chosen from the first 11 digits. Let us look at these first 11 digits: $12345123451$ The largest digit in this part is 5. We choose the first '5' we find. This '5' is the 5th digit of the original number. We strike out the digits before it. $1, 2, 3, 4$ The number of digits struck out is 4. The first digit of our new number is 5. The remaining number starts after this '5'. It is $123451234512345$ . It has $20 - 5 = 15$ digits.

Step 2 — Finding the second digit

We need to choose the second digit. We have 9 digits left to choose. The remaining part of the original number has 15 digits. So, the second digit must be chosen from the first $15 - 8$ digits. $15 - 8 = 7$ So, the second digit must be chosen from the first 7 digits. Let us look at these first 7 digits: $1234512$ The largest digit in this part is 5. We choose the first '5' we find. This '5' is the 5th digit of this remaining part. We strike out the digits before it. $1, 2, 3, 4$ The number of digits struck out is 4. Total digits struck out so far: $4 + 4$

$\boxed{8 \text{ digits}}$ The second digit of our new number is 5. The remaining number starts after this '5'. It is $1234512345$ . It has $15 - 5 = 10$ digits.

Step 3 — Finding the third digit

We need to choose the third digit. We have 8 digits left to choose. The remaining part of the original number has 10 digits. So, the third digit must be chosen from the first $10 - 7$ digits. $10 - 7 = 3$ So, the third digit must be chosen from the first 3 digits. Let us look at these first 3 digits: $123$ The largest digit in this part is 3. We choose this '3'. This '3' is the 3rd digit of this remaining part. We strike out the digits before it. $1, 2$ The number of digits struck out is 2. Total digits struck out so far: $8 + 2$

$\boxed{10 \text{ digits}}$ The third digit of our new number is 3. The remaining number starts after this '3'. It is $4512345$ . It has $10 - 3 = 7$ digits.

Step 4 — Completing the number

We have now struck out exactly 10 digits. We have chosen 3 digits for our new number. These digits are 5, 5, and 3. We need to choose $10 - 3 = 7$ more digits. The remaining part of the original number is '4512345'. This part has exactly 7 digits. We have struck out 10 digits. So, we must keep all remaining digits. The remaining digits are 4, 5, 1, 2, 3, 4, 5. We append these to our chosen digits. The final number is formed by combining these digits. $5534512345$

Answer

The largest possible number is 5534512345.

Question 16

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