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How many different integers can be represented as a sum of four distinct numbers chosen from the set {5,12,19,26,..., 110} ?
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The given set represents the terms of the arithmetic progression with the first term of 5 and the common difference of 7:
= 5 + 7m, m = 0, 1, 2, 3, . . . , 15.
In all, the set has 16 elements.
The sum of any 4 numbers of the set is the number of the form ,
where , , and are different integer numbers between 0 and 16, inclusive.
The minimum value of such sum is, OBVIOUSLY, 0+1+2+3 = 6.
The maximum value of such sum is, OBVIOUSLY, 12+13+14+15 = 54.
It is clear that any index from 6 to 54 can be obtained as the sum of this form .
THEREFORE, the ANSWER to the problem's question is 54 - 5 = 49.
ANSWER. 49 different integers can be represented as the sum.
Solved.