From the set {1, 2, 3, 4, 5, 6}, in how many ways can three distinct numbers be selected so that they can form the sides of a non-degenerate triangle? (Order does not matter; {3,4,5} is the same as {4,3,5}.)

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7. Total triples: C(6,3) = 20. Keep those where the two smaller sides sum to more than the largest. Valid sets: {2,3,4},{2,4,5},{2,5,6},{3,4,5},{3,4,6},{3,5,6},{4,5,6}. That is 7 sets. (Checks: {2,3,4}: 2+3>4 yes; {2,4,5}: 2+4>5 yes; {2,5,6}: 2+5>6 yes; {3,4,5}: 7>5; {3,4,6}: 7>6; {3,5,6}: 8>6; {4,5,6}: 9>6.) Total = 7.

From the set {1, 2, 3, 4, 5, 6}, in how many ways can three distinct numbers be selected so that they can form the sides of a non-degenerate triangle? (Order does not matter; {3,4,5} is the same as {4,3,5}.)

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