A probability problem by Paola Ramírez

Each number from 1 to $10^{2000}$ consists of 2000 digits (combination of digits 0,1,2,3,4,5,6,7,8,9)

Getting the no. of numbers whose sum of digits is equal to 2, is equivalent to getting the coefficient of $x^{2}$ in $(x^{0}+ x^{1}+ x^{2}+ x^{3}+ x^{4}+ x^{5}+ x^{6}+ x^{7}+ x^{8}+ x^{9} ) ^{2000}$

$(x^{0}+ x^{1}+ x^{2}+ x^{3}+ x^{4}+ x^{5}+ x^{6}+ x^{7}+ x^{8}+ x^{9} ) ^{2000}$ * $(x-1)^{2000}$ * $\frac{1}{(x-1)}^{2000}$

= $(1-x^{10})^{2000}$ * (n+1999) C n * $x^{n}$

Expanding $(1-x^{10})^{2000}$ , we'll get terms with exponent of x greater than 2, so we disregard these terms.

= 1 * (n+1999) C n * $x^{n}$

Since we are looking for the coefficient of $x^{2}$ , then n must be equal to 2

The coefficient of $x^{2}$ is 2001 C 2 = $\boxed{2001000}$

A quick observation:- No. of single digit number having above property :- (obviously) 1 i.e. the number 2; No of 2 digit number :- 2 (11, 20) No. of 3 digit number :- 3 (101, 110, 200) Clearly the solution of n digit number can be given as no. permutation of $10^{n - 1} + 1$ keeping first digit $(1)$ fixed and plus 1 for $2 \cdot 10^{n - 1}$ . Ex :- for 3 digit number, number of permutation of $10^2 +1 = 101$ , keeping first digit $(1)$ fixed, is 2 and add 1 for 200. Now clearly the number of solution for n digit number is $\boxed{n}$

$\therefore$ our ans is simple $\dfrac{2000 \cdot 2001}{2} = \boxed{2001000}$

Given the 10 numbers up to $10^4$ that follow the rule:

We can see that there's 1 for a single digit, 2 for a pair of them, 3 for three and linearly increasing. So $10^{2000}$ means the result must be a sumattory from 1 to 2000, which becomes easy if we consider 1000 times 2001.

(Gauss saw that, for our example of $10^4$ : $1+2+3+4 = (1+4)+(2+3) = 5+5 = 2 \cdot 5 = \frac {4 \cdot 5}{2} = 10$ .)

So $\frac {2000 \cdot 2001}{2} = \boxed{2001000}$

We observe that any such number satisfying the above conditions are of the form $\overline{a_1a_2...a_{2001}},\quad a_1,a_2,...,a_{2001}\geq0\quad ,a_1+a_2+...a_{2001}=2$ By stars and bars, the answer is 2001 choose 2, which is 2001000.