Nancy is asked to select integers, between 0 and 10^13. If shes instructed to select only integers whose sum of digits is 3, then how many different integers can she select within the given range?

Question

Simran Mehta · Accepted Answer

To solve this, we need to find how many different integers between 0 and 10^13 have a digit sum of 3.
Let's consider the digits of a 13-digit number (including leading zeros). We need to find non-negative integer solutions to the equation:
d1 + d2 + ... + d13 = 3where 0 &le; di &le; 9 for each i.
This is a classic "stars and bars" problem in combinatorics, where we are distributing 3 "stars" (the sum) into 13 "bins" (the digits).
The number of solutions is given by the binomial coefficient:C(n + k - 1, k - 1)where n is the sum (3) and k is the number of bins (13).
So we need to calculate:C(3 + 13 - 1, 13 - 1) = C(15, 12)
C(15, 12) is the same as C(15, 3), due to the symmetry of binomial coefficients:C(15, 3) = 15! / (3!(15 - 3)!)
= 15! / (3!12!)
= (15 * 14 * 13) / (3 * 2 * 1)= 455
Thus, Nancy can select 455 different integers within the given range.

Nancy is asked to select integers, between 0 and 10^13. If she's instructed to select only integers whose sum of digits is 3, then how many different integers can she select within the given range?