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No big digits involved (Posted on 2017-11-06) Difficulty: 3 of 5
How many decimal integers below 10^20 exist having sum of their digits less than 4?

No Solution Yet Submitted by Ady TZIDON    
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Solution solution Comment 2 of 2 |
We can consider 20-digit numbers with leading zeros allowed--just lop off any leading zeros for the final result, as there's still a 1-to-1 correspondence of numbers with leading zeros and a set of numbers with leading blanks.

Now if we know what the possible combinations of non-zero digits are, we can arrange them in the various positions.

The non-zero digits could be:

11, 2
111, 12, 3

These are the 6 possibilities. How many placement possibilities are for each of these combinatinos?

1:   20
11:  C(20,2)= 190
2:   20
111: C(20,3)= 1140
12:  P(20,2)= 380
3:   20

20+190+20+1140+380+20 = 1770

If we count zero, we can add 1 to the total.

  Posted by Charlie on 2017-11-06 18:04:58
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