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