Pre-RMO 2014/20

What is the number of ordered pairs (A,B)(A, B) where AA and BB are subsets of {1,2,,5}\{1, 2, \ldots, 5\} such that neither ABA \subseteq B nor BAB \subseteq A ?


This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
4 years, 8 months ago

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is it 30???

Nitish Deshpande - 4 years, 8 months ago

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How many elements can A or B can have? Only 1 or more than one

If only 1 element ans \to 20

Otherwise(counting every possibility) ans \to 47

Krishna Sharma - 4 years, 8 months ago

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Since AA and BB are subsets of {1,2,3,4,5}\{1, 2, 3, 4, 5\}, they can have from 0 to 5 elements.

Pranshu Gaba - 4 years, 8 months ago

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Write a comment or ask a question... I think 52

vatsal shah - 4 years, 8 months ago

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should be 1024 - 243 - 243 + 32 = 570

ww margera - 4 years, 8 months ago

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ww margera. actually you permuted it...... {1,2} is Same as {2,1}

vatsal shah - 4 years, 8 months ago

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Ordered pairs right? So the two should not be the same...

ww margera - 4 years, 8 months ago

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@ww margera. actually you permuted it...... {1,2} is Same as {2,1}

vatsal shah - 4 years, 8 months ago

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As @ww margera said, we can use principle of inclusion and exclusion.

Answer=total no. of ordered pairs(AB+BA)+A=B \text{Answer} = \text{total no. of ordered pairs} - (|A \subseteq B |+ |B \subseteq A|) + |A = B|

where x|x| is no. of ordered pairs (A,B)(A, B) satifying condition xx.

Answer=210(35+35)+25=570 \text{Answer}= 2^{10} - ( 3^5 + 3^5) + 2^5 = \boxed{570}

Pranshu Gaba - 4 years, 8 months ago

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How 2^10, 3^5, 2^5, I didn't get that.

Raj Hariya - 10 months, 3 weeks ago

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why should it be 2^10......i think it should be ${2^5 \choose 2}*2$

Abhishek Bakshi - 4 years, 8 months ago

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Using rule of product, AA has 252^5 choices, and even BB has 252^5 choices, so total no. of ordered pairs = 25×25=2102^5 \times 2^5 = 2^{10}

Pranshu Gaba - 4 years, 8 months ago

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6

Muhammad Adil Zafar - 4 years, 8 months ago

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570

Saarthak Marathe - 3 years, 8 months ago

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@Pranshu Gaba : Are you studying in resonance? Which Centre?

Abhishek Bakshi - 3 years, 8 months ago

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