Question 1099274: Given the set {1,1,2,4,7,13,24,44,81}, how many integers between 3 and 85 cannot be written as the sum of three elements in the set?
Found 2 solutions by Boreal, greenestamps:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The integers that aren't able are
81+44+(1,2,4,7,13,24)---6 of them
81+24+(1,2,4,7,13,44)-- 6 more, but 81+24+44 is double counted, so 5 more
81+13+(1,2,4,7,24,44)--6 more but 81+13+24 is double counted, as is 44 so 4
81+7+(1,2,4)--3 more
81+4+(1,2)--2 more
20 integers not able
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
I'm not sure what the other tutor was doing; the work he shows has nothing to do with the question that was asked....
I doubt that there is a fancy mathematical method that will get you to the answer any faster than simply identifying which numbers between 3 and 85 can and cannot be formed as the sum of three elements of the given set. I would just use a logical process to find out which sums can be made with three elements of the set....
if the two smallest elements are 1 and 1: {4, 6, 9, 15, 26, 46, 83}
if the two smallest elements are 1 and 2: {7, 10, 16, 27, 47, 84}
if the two smallest elements are 1 and 4: {12, 18, 29, 49}
if the two smallest elements are 1 and 7: {21, 32, 52}
if the two smallest elements are 1 and 13: {38, 58}
if the two smallest elements are 1 and 24: {69}
if the two smallest elements are 2 and 4: {13, 19, 30, 50}
if the two smallest elements are 2 and 7: {22, 33, 53}
if the two smallest elements are 2 and 13: {39, 59}
if the two smallest elements are 2 and 24: {70}
if the two smallest elements are 4 and 7: {24, 35, 55}
if the two smallest elements are 4 and 13: {41, 61}
if the two smallest elements are 4 and 24: {72}
if the two smallest elements are 7 and 13: {44, 64}
if the two smallest elements are 7 and 24: {75}
if the two smallest elements are 13 and 24: {81}
It turns out no numbers between 3 and 85 can be made using three numbers from the set in two different ways. The total number of numbers between 3 and 85 that CAN be formed as the sum of three numbers from the set is 43 (by counting the elements of all the sets above).
Taking "between 3 and 85" to mean not counting either 3 or 85, there are 81 numbers between 3 and 85; since 43 of them CAN be formed by the sum of three elements of the given set, the answer to the problem -- the number of numbers between 3 and 85 that CANNOT be made as the sum of three numbers of the given set -- is 81-43 = 38.
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