SOLUTION: Maria has seven cards with the digits 1 to 7 written on them, as shown. 1,2,3,4,5,6,7 She knows that a number is divisible by nine if the sum of its digits is divisible by nine.

Question 629895: Maria has seven cards with the digits 1 to 7 written on them, as shown.
1,2,3,4,5,6,7
She knows that a number is divisible by nine if the sum of its digits is divisible by nine.
How many three-digit numbers that are divisible by nine can Maria make using any three of her cards?
Answer by Charles3475(23) (Show Source): You can put this solution on YOUR website!
the smallest sum divisible by 9 would be 9. How many ways can we use three cards to get 9?

1 + 6 + 2 = 9 (so 162)
1 + 5 + 3 = 9 (so 153)
1 + 3 + 5 = 9 (so 135)
1 + 2 + 6 = 9 (so 126)
2 + 6 + 1 = 9 (so 261)
2 + 4 + 3 = 9 (so 243)
2 + 3 + 4 = 9 (so 234)
2 + 1 + 6 = 9 (so 216)
3 + 5 + 1 = 9 (so 351)
3 + 4 + 2 = 9 (so 342)
3 + 2 + 4 = 9 (so 324)
3 + 1 + 5 = 9 (so 315)
4 + 3 + 2 = 9 (so 432)
4 + 2 + 3 = 9 (so 423)
5 + 3 + 1 = 9 (so 531)
5 + 1 + 3 = 9 (so 513)
6 + 2 + 1 = 9 (so 621)
6 + 1 + 2 = 9 (so 612)

The next sum divisible by 9 would be 18. How many ways can we use three cards to get to 18?

7 + 6 + 5 = 18 (so 765)
7 + 5 + 6 = 18 (so 756)
6 + 7 + 5 = 18 (so 675)
6 + 5 + 7 = 18 (so 657)
5 + 7 + 6 = 18 (so 576)
5 + 6 + 7 = 18 (so 567)

So I get a total of 24 ways
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