Permutations and Combinations Part 1

For odd numbers, unit place shall be 1,3,5,7 or 9 .
xxyxy,xxyxy,xxyxy,xxyxz,xxyx9 are the type of numbers.
If xy1 then
x+y=6,13,20⋯
Total number =6+6+0+⋯=12
If x y 3 then
x+y=4,11,18,⋯
Total number =4+8+1+0⋯=13
Similarly for x y 5 , we have
x+y=2,9,16,⋯
Total number =2+9+3=14
for x y 7 we have
x+y=0,7,14,⋯
Total number =0+7+5=12 ways
And for x y 9 we have
x+y=5,12,19...
Total number =5+7+0⋯=12 ways
∴ Total odd numbers whose sum of digits is a multiple of 7 is 63 .