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Combinatorics-RMO 2015

Suppose there are 36 equally spaced points on the circumference of a circle.In how many ways you can choose 3 any two points such that no 2 of them are adjacent or diametrically opposite?

Note by Souryajit Roy
9 months, 4 weeks ago

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We shall use PIE.

Step 1: Number of ways of selecting \(3\) points from \(36\) points is \(\binom{36}{3} = 7140\).

Step 2: Number of ways of selecting \(3\) adjacent points is \(36\).

Step 3: Number of ways of selecting \(2\) adjacent and one not adjacent with them is \(36\times 32 = 1152\). (Since there are \(32\) ways to select the non-adjacent point.)

Step 4: Number of ways of selecting two diametrically opposite points are \(18\) and number of ways of selecting third one not adjacent to both of them are \(30\) in each case. So total number of ways in this step are \(18 \times 30 = 540\).

Step 5: Number of ways (what we required) = Total \(-\) Number of ways of selecting \(3\) adjacent points \(-\) Number of ways of selecting \(2\) adjacent and one not adjacent with them - Number of ways of selecting two diametrically opposite points and selecting third one not adjacent to both of them \(= 7140-36-1152-540=5412\). Surya Prakash · 9 months, 4 weeks ago

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@Surya Prakash no of ways of selecting two adjancent and the third being diametrically opposite to one of the two is : 36 * 2 = 72 , I too gave the rmo this year of west bengal. Aditya Sharma · 9 months, 4 weeks ago

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I think there were 32 points instead of 36. Prakher Gaushal · 9 months, 4 weeks ago

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@Prakher Gaushal I have given RMO of West Bengal Rgion and there was 36 points in the problem (problem no 4)....Perhaps in your region they have given 32....basically the procedure of solving the sum is not affected if the number of points in even. Souryajit Roy · 9 months, 4 weeks ago

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The answer is 5400.

Let the points be A,B and C. We can put A anywhere on the circle, giving 36 ways.

Then, B and C cannot be on the two points next to A, or the point opposite it, leaving 32C2 = 496 ways to choose B and C.

Of these, 30 configurations have B and C adjacent, and 16 have them opposite. We subtract these off to give 450 valid ways.

For each valid configuration, there is one way to generate it starting with A at each vertex. Thus each configuration has been counted 3 times, so we divide by 3 to get the final answer:

36x450/3 = 5400 Daniel Tan · 8 months, 1 week ago

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What can be expected cutoff for rmo Satyam Mani · 9 months, 3 weeks ago

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@Satyam Mani around 55 Raushan Sharma · 9 months, 3 weeks ago

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For 32 objects you can have a look at my solution here Shubhendra Singh · 9 months, 4 weeks ago

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5430 Prakher Gaushal · 9 months, 4 weeks ago

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32*36 Dipanjan Chowdhury · 9 months, 4 weeks ago

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What was your answer? Prakher Gaushal · 9 months, 4 weeks ago

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