Question

How many integers between 1 and 10^21 are such that the sum of their digits is 2?

Answer 1

Nice Q....
So, we're about to find "How many integers between 1 and 10^21 are such that the sum of their digits is 2"

For No. of digits =  1, no. of such no.s = 1, i.e., 2
For No. of digits =  2, no. of such no.s = 2, i.e., 11, 20
For No. of digits =  3, no. of such no.s = 3, i.e., 101,110,200
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Till, for  No. of digits = n , no. of such no.s = n

Hence, between 1 and 10^21, there is $S_{21}$ no.s 
= 21/2 { 22 } = 21*11 = 231 such no.s