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Sum of the Digits 7

How many natural numbers less that \(10^8 \) are there, whose sum of digits equals 7?

Solution:

Consider the equation \(x_1 + x_2 + … + x_8 = 7 \)

Number of non negative integer solutions to this equation is \(\dbinom{14}{7} \)

The set of solutions is bijective with the number of numbers less that \(10^8 \) whose digit adds up to 7. Hence the answer is \(\dbinom{14}{7} \)

Key Idea: Bijection; Number of Non Negative Integer Solutions to an equation

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