Lottery also comes in numbers!
A lottery comes in form of 6-digit numbers from
to
.
Let us call a lottery lucky if the sum of the first 3 digits are equal to the last 3 digits. For example, is lucky because .
And let us call a lottery good if the sum of all digits is 27. For example, is good because .
If the number of different possible lotteries that are lucky is , and the number of different possible lotteries that are good is , find the value of .
This question is from Thailand Math POSN, Combinatorics.
The answer is 0.
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1 solution
Let L be the set of lucky numbers and G be the set of good numbers. We want to prove that there is a bijection between the two sets.
First, we see that for any lucky number a b c d e f with a + b + c = d + e + f , if we replace d , e , f with 9 − d , 9 − e , 9 − f , we get a good number. We also see that no two lucky numbers can give the same good number, as it would mean that they are the same.
Likewise, for any good number a b c d e f , we can get a lucky number by replacing d , e , f with 9 − d , 9 − e , 9 − f . Also, as above, no two good numbers can give the same lucky number.
Therefore, a bijection exists between the sets and a − b = 0 .