How many 10 digit numbers are there so the sum of the digits is $2$?
$abcdefghij$ is the 10 digit number. By default, $a=1$ is a must.
$= 1bcdefghij$
Now we need: $bcdefghij = 1$
How can I solve this combinatorically? Not by checking and substitution?
Best Answer
if $a=1$, for the rest $9$ places, $1$ can be taken anywhere, and also the rest $8$ places must be $0$ or the sum would exceed $2$.
Also if $a=2$, then the rest $9$ places must be $0$, or sum would exceed $2$.
Hence there are $9+1=10$ such numbers.