Question 7
All the positive integers whose sum of digits is 7 are written in the increasing order. The first few are $$7, 16, 25, 34, 43, \ldots$$ What is the 125 th number in this list?
Solution
The numbers below 10000 with digit sum 7 correspond to solutions of $$d_1+d_2+d_3+d_4 = 7$$ with each digit at most 9, and there are $$\binom{10}{3} = 120$$ of them. So the list continues with the five digit numbers $$10006, 10015, 10024, 10033, 10042$$ as the 121st to 125th entries. Hence the 125th number is 10042.
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