Lottery also comes in numbers!

A lottery comes in form of 6-digit numbers from \(000000\) to \(999999\).

Let us call a lottery lucky if the sum of the first 3 digits are equal to the last 3 digits. For example, \(456177\) is lucky because \(4+5+6 = 1+7+7\).

And let us call a lottery good if the sum of all digits is 27. For example, \(345456\) is good because \(3+4+5+4+5+6 = 27\).

If the number of different possible lotteries that are lucky is \(a\), and the number of different possible lotteries that are good is \(b\), find the value of \(a-b\).


This question is from Thailand Math POSN, Combinatorics.

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