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# Pre-RMO 2014/20

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

This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
2 years, 7 months ago

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@Pranshu Gaba : Are you studying in resonance? Which Centre? · 1 year, 7 months ago

570 · 1 year, 7 months ago

6 · 2 years, 7 months ago

why should it be 2^10......i think it should be ${2^5 \choose 2}*2$ · 2 years, 7 months ago

Using rule of product, $$A$$ has $$2^5$$ choices, and even $$B$$ has $$2^5$$ choices, so total no. of ordered pairs = $$2^5 \times 2^5 = 2^{10}$$ · 2 years, 7 months ago

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

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

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

$\text{Answer}= 2^{10} - ( 3^5 + 3^5) + 2^5 = \boxed{570}$ · 2 years, 7 months ago

@ww margera. actually you permuted it...... {1,2} is Same as {2,1} · 2 years, 7 months ago

should be 1024 - 243 - 243 + 32 = 570 · 2 years, 7 months ago

ww margera. actually you permuted it...... {1,2} is Same as {2,1} · 2 years, 7 months ago

Ordered pairs right? So the two should not be the same... · 2 years, 7 months ago

Write a comment or ask a question... I think 52 · 2 years, 7 months ago

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 · 2 years, 7 months ago

Since $$A$$ and $$B$$ are subsets of $$\{1, 2, 3, 4, 5\}$$, they can have from 0 to 5 elements. · 2 years, 7 months ago