Mathcounts National 2014 Sprint #15

Question

This problem is from the 2014 Mathcounts National Sprint test. The correct answer is 13/1024, but I keep getting 17/2048s.

My attempt at this problem was to use casework. Obviously, the denominator of this fraction will come out to 161616 = 2^12 = 4096. For the numerator, I came up with 4 cases for how the sum of the squares could be over 15.

1. 7 7 7 :There is one order that this can be thrown in and all three of them must be from the same 1 square so this combination gives us 1.

2. 7 7 5 : There is 3 ways this can be ordered * 3 squares the five can be picked from to give 9.

3. 7 7 3: 3 orders possible * 5 squares for the three so 15.

4. 7 5 5: 3 orders possible * (3 squares choose 2 fives) = 3 * 3 = 9.

In total, from these cases, we have 9 + 9 + 15 + 1 = 34.

34/2^12 = 17/2^11 = 17/2048.

Am I missing cases? Or did I calculate the number of ways incorrectly. Please let me know of any edits I should make to the question before you downvote it.

EDIT: I just realized that the way I calculated case 4 was incorrect, as I assumed that the five squares needed to be 2 different squares. In reality, there are 331 total ways to choose squares times the 3 orders = 27. This makes the final result 52/2^12 = 13/1024.

Answer 1

The mistake I had made was that for case #4 "7 5 5", I used 3 choose 2 to determine how many ways the squares could be chosen for the fives. However, I did not take into account that the squares could be the same. This would mean that rather than 3 choose 2, it would be 3 * 3, as there are 3 squares for the first 5 and 3 squares for the second 5. 9 * the 3 ways to order the case give 27, not 9.