In how many ways can the three chosen rods form a triangle?
A hard-level "Count" problem.Whether three lengths make a triangle rests on one test: the longest must be less than the other two added. Equal collapses flat, so it does not count.
GivenThere is one rod of each length from 1 cm to 6 cm. Three of them are chosen.
Hints
- Three rods do not always make a triangle. Try 1, 2 and 4 cm: the two short ones stretch to only 1 + 2 = 3 cm, which never reaches the end of the 4 cm rod.
- What about 1, 3 and 4? Now 1 + 3 = 4 reaches exactly. But exactly is no good — the shape collapses flat and is not a triangle. Equal fails too.
- So there is just one rule: the two shorter rods together must be longer than the longest one. With 2, 3, 4: 2 + 3 = 5 > 4, with room to spare — a genuine triangle.
- Now just check them. Choosing 3 rods from 6 gives 6 × 5 × 4 ÷ (3 × 2 × 1) = 20 ways. Sift those 20 through the rule we just made.
- The survivors are 2-3-4, 2-4-5, 2-5-6, 3-4-5, 3-4-6, 3-5-6, 4-5-6 — 7 ways. Fix the longest rod first and then choose the other two: that way nothing is missed.
Answer 7
More "Count" problems
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- How many cubes have 2 painted faces?
- Into how many pieces do 4 lines cut the paper?
- How many triangles are there in the figure?
- How many squares are there altogether?
- How many squares does the diagonal pass through?