Problem 14 · AMC 8 Stretch
Stretch
Counting & Probability
Geometry & Measurement
and-process-multiplylogical-reasoning
In how many ways can a triangle be named using 3 different letters of the 26-letter alphabet? (The same triangle can be read starting at any of its 3 corners and in either direction, so different readings of the same triangle should NOT be counted as different.)
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Answer: 2600 triangles
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Hint 1 of 4
First count ORDERED lists of 3 different letters (an AND process: 26 then 25 then 24). Then fix the over-count.
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Hint 2 of 4
The triangle named ABC is the same triangle as BCA, CAB (different starting corners) and also as the reverse readings ACB, CBA, BAC.
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Hint 3 of 4
Each triangle gets counted 6 times: 3 starting corners times 2 directions. So divide the ordered count by 6.
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Approach: Count ordered lists, then divide out the repeats
- A triangle's name lists its 3 corners, but the SAME triangle can start at any of the 3 corners and go either of 2 directions. So first count ordered lists, then divide out these repeats.
- Ordered lists of 3 different letters: \(26 \times 25 \times 24 = 15600\).
- Each triangle is named \(3 \times 2 = 6\) different ways, so divide: \(\dfrac{15600}{6} = 2600\).
- (This is the same as 'just choose which 3 letters', since once you pick 3 corners there is only one triangle.)
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