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# counting problem

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Find the number of 10-digit numbers, where the sum of the digits is divisible by $$15$$

May 14, 2022

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So, suppose the digits were $$a_1, a_2, \cdots , a_{10}$$ and a_1 is the first digit, a_2 is the second digit, etc.

Note that the first digit can't be 0. So the required number is

$$(\#\text{ of cases where }a_1 + a_2+ \cdots + a_{10} = 15) - (\#\text{ of cases where }a_2 + a_3 + \cdots + a_{10} = 15)$$

Each number can be calculated using this theorem described on this wiki page: https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)#Theorem_two.

May 14, 2022