# 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