Number of lattice points ( x , y , z all being integers) inside the tetrahedron (not on the surface) having vertices ( 0 , 0 , 0 ), (21, 0, 0), (0, 21, 0), (0, 0, 21) is

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The tetrahedron is bounded by x ≥ 0 , y ≥ 0 , z ≥ 0 and x + y + z = 21 . Interior lattice points have x , y , z ≥ 1 and x + y + z ≤ 20 . Their number is ( 3 20 ​ ) = 1140 .

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