There are 1680 digits 2, 0, 2, and 2 as a 4-digit consecutively ordered block with no other digits between them
How to determine the selection
To determine the number of whole numbers, the following must be true
Case 1: If the sequence starts from the first digit
- The first digit can be any of the three 2's (i.e. 3 digits)
- The second digit can only be 0
- The third digit can be any of the remaining 2's (i.e. 2 digits)
- The fourth digit can only be the last 2 (i.e. 1 digit)
- The fifth digit can be any of 0 - 9 (i.e. 10 digits)
- The sixth digit can be any of 0 - 9 (i.e. 10 digits)
So, we have:
[tex]Case\ 1 = 3 * 1 * 2 * 1 * 10 * 10[/tex]
[tex]Case\ 1 = 600[/tex]
Case 2: If the sequence starts from the second digit
- The first digit can be any of 1 - 9 (i.e. 9 digits)
- The second digit can be any of the three 2's (i.e. 3 digits)
- The third digit can only be 0
- The fourth digit can be any of the remaining 2's (i.e. 2 digits)
- The fifth digit can only be the last 2 (i.e. 1 digit)
- The last digit can be any of 0 - 9 (i.e. 10 digits)
So, we have:
[tex]Case\ 2 = 9 * 3 * 1 * 2 * 1 * 10[/tex]
[tex]Case\ 2 = 540[/tex]
Case 2: If the sequence starts from the third digit
- The first digit can be any of 1 - 9 (i.e. 9 digits)
- The second digit can be any of 0 - 9 (i.e. 10 digits)
- The third digit can be any of the three 2's (i.e. 3 digits)
- The fourth digit can only be 0
- The fifth digit can be any of the remaining 2's (i.e. 2 digits)
- The last digit can only be the last 2 (i.e. 1 digit)
So, we have:
[tex]Case\ 3 = 9 * 10 * 3 * 1 * 2 * 1[/tex]
[tex]Case\ 3 = 540[/tex]
The total number of whole numbers is:
[tex]Total = 600 + 540 + 540[/tex]
[tex]Total = 1680[/tex]
Hence, there are 1680 6-digit whole numbers that the digits
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