Problem 17 · 2010 Math Kangaroo
Hard
Counting & Probability
careful-countingsymmetry
Lydia draws a flower with 5 petals. She wants to colour the petals using only the colours white and black. Two flowers count as the same if one can be turned to look like the other. How many different flowers can she make (the flower may also be just one colour)?
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Answer: C — 8
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Hint 1 of 2
Turning a flower around does not make a new flower, so group your counting by how many petals are black.
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Hint 2 of 2
For 2 black petals, the only thing that matters is whether the two black petals touch or have a gap between them.
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Approach: sort flowers by how many black petals
- Count by the number of black petals, since spinning the flower never makes a new one.
- 0 black: 1 way. 1 black: 1 way. 2 black: 2 ways (the black petals touch, or have a gap). 3 black: 2 ways. 4 black: 1 way. 5 black: 1 way.
- Adding up: \(1 + 1 + 2 + 2 + 1 + 1 = 8\) different flowers — the answer is C.
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