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

#Combinatorics #Pre-RMO

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

is it 30???

Nitish Deshpande - 6 years, 8 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

Krishna Sharma - 6 years, 8 months ago

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

Pranshu Gaba - 6 years, 8 months ago

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

vatsal shah - 6 years, 8 months ago

should be 1024 - 243 - 243 + 32 = 570

ww margera - 6 years, 8 months ago

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

vatsal shah - 6 years, 8 months ago

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

ww margera - 6 years, 8 months ago

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

vatsal shah - 6 years, 8 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 $x$ .

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

Pranshu Gaba - 6 years, 8 months ago

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

Raj Hariya - 2 years, 10 months ago

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

Abhishek Bakshi - 6 years, 8 months ago

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

Pranshu Gaba - 6 years, 8 months ago

Muhammad Adil Zafar - 6 years, 8 months ago

570

Saarthak Marathe - 5 years, 8 months ago

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

Abhishek Bakshi - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$