How many three digit numbers are there whose first digit equals the su

In such questions, it is good to write down the ways as it does not entail too much of work...
Let the number be xyz
1) x is 9, so y+z=9
Ways : (9,0); (8,1);(7,2);(6,3);(5,4) ..
Each of the above 5 ways can be taken as 2 different arrangements.. 981 or 918
So 5*2=10
2) x is 8, so y+z=8
Ways : (8,0);(7,1);(6,2);(5,3);(4,4) ..
4 of the above 5 ways can be taken as 2 different arrangements..
So 4*2+1=9
3) x is 7, so y+z=7
Ways : (7,0);(6,1);(5,2);(4,3)
Each of the above 4 ways can be taken as 2 different arrangements..
So 4*2
4) x is 6, so y+z=6
Ways : (6,0);(5,1);(4,2);(3,3)....
3 of the above 4 ways can be taken as 2 different arrangements..
So 3*2+1=7
And so on
There is a pattern...
So 10+9+8+7+6+5+4+3+2=54
Last 2 is when x is 1...
101 or 110

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