Question 6
How many three-digit positive integers are there if the digits are the side lengths of some isosceles or equilateral triangle?
Solution
No digit can be 0, and at least two digits must be equal. Equilateral cases give 9 numbers. For a pair $$(x, x, y)$$ with $$y \neq x$$ the triangle inequality needs $$y < 2x$$, and counting valid $$y$$ for $$x = 1$$ to $$9$$ gives $$0+2+4+6+8+8+8+8+8 = 52$$ multisets, each arrangeable in 3 ways. The total is $$9 + 3 \times 52 = 165$$.
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