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Ways of taking out numbers

Source: 1989 National High School Mathematics League, Exam One, Problem 11

February 25, 2020

Problem Statement

From 1,2,⋯ ,141,2,\cdots,14, take out three numbers a1<a2<a3a_1<a_2<a_3, satisfying that a2−a1≥3,a3−a2≥3a_2-a_1\geq3,a_3-a_2\geq3. Then the number of different ways of taking out numbers is________.
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