Possible solutions in range for Inequality with multiple variables

Questions
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1

So this might be complicated but I am trying to find the proper command .
I need the possible solutions for

119< 5 x1 + 10 x2 + 7 x3 + 5.2 x4  + 6.7 x5 <120

with

x1, x2, x3, x4, x5 ∈ [1 ..5]

I already tried

NSolve [{
  119< 5 x1 + 10 x2 + 7 x3 + 5.2 x4  + 6.7 x5 <120,
  Element[ {x1, x2, x3, x4, x5}, Point @ { {0},{1},{2},{3},{4} } ] }, { x1, x2, x3, x4, x5 }]

But somehow it doesn’t compute? Is there another maybe more elegant way?

2

Rob,

here is one simple way:

maxSols = 400;
(* get 'maxSols' solutions: *)
sols = FindInstance[119 < 5 x1 + 10 x2 + 7 x3 + 5.2 x4 + 6.7 x5 < 120, {x1, x2, x3, x4, x5}, PositiveIntegers, maxSols];
(* if "True" we might have found all solutions: *)
Length[sols] < maxSols

and then:

(* select solutions in the respective interval: *)
Select[sols, x1 <= 5 && x2 <= 5 && x3 <= 5 && x4 <= 5 && x5 <= 5 /. # &]
3

Another way:

Solve[119 < 5 x1 + 10 x2 + 7 x3 + 52/10 x4 + 67/10 x5 < 120 && 
  0 <= x1 <= 5 && 0 <= x2 <= 5 && 0 <= x3 <= 5 && 0 <= x4 <= 5 && 
  0 <= x5 <= 5, {x1, x2, x3, x4, x5}, Integers]