Question 1208788
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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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<pre>
The given set represents the terms of the arithmetic progression with the first term of 5 and the common difference of 7:

    {{{a[n]}}} = 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  {{{20 + 4(m[1]+m[2]+m[3]+m[4])}}} ,

where  {{{m[1]}}}, {{{m[2]}}}, {{{m[3]}}}  and  {{{m[4]}}}  are different integer numbers between 0 and 16, inclusive.


The minimum value of such sum  {{{m[1]+m[2]+m[3]+m[4]}}}  is, OBVIOUSLY,  0+1+2+3 = 6.


The maximum value of such sum  {{{m[1]+m[2]+m[3]+m[4]}}}  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  {{{m[1]+m[2]+m[3]+m[4]}}}.


THEREFORE, the  <U>ANSWER</U> to the problem's question is  54 - 5 = 49.


<U>ANSWER</U>.  49 different integers can be represented as the sum.
</pre>

Solved.