How many 10 digit numbers are there so the sum of the digits is $2$?

Question

$abcdefghij$ 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?

"checking and substitution" opens quite often a fruitful route to "combinatorics". — drhab, Jan 26, 2015 at 18:03

Shobhit · Accepted Answer · 2015-01-26 18:05:13Z

6

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.

answered Jan 26, 2015 at 17:59

Shobhit

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$\begingroup$ I assume you meant that the rest of the $9$ places would be $0$ if $a = 2$. $\endgroup$
– N. F. Taussig
Jan 26, 2015 at 18:04
$\begingroup$ @N.F.Taussig yes ofcourse, thank you :) $\endgroup$
– Shobhit
Jan 26, 2015 at 18:05

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