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In school I learnt to create a set of number or how many sets of number we can create from given numbers. But now, I forgot. Please help me here.

I want to know how many sets can be possible to create and I want create 2 number in 1 set from 1 to 10

Example : Number are 1 to 10 Sets should be like (1,1), (1,2), (1,3), ....................(5,1), (5,2), ....................(10,8), (10,9), (10,10).

Thank you for your help.

[EDITED]

As I remember it can be possible like aª. And a=10 in my case. So the answer will be aª=10×10=100.

Bel
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 25 '22 at 04:17
  • It is not clear what you are asking. Since you have (I think) the elements $1,...,10$, a set can be formed from any subset of these. So $1$ may or may not be in the set, $2$ may or may not be in the set, etc. Hence there are $2^{10}$ possible sets, ranging from the empty $\emptyset = {}$ all the way to ${1,...,10}$. – copper.hat Oct 25 '22 at 04:23
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    Note that $a^a = 10^{10} = 1 0000000000$ is quite different from $a^2 = 10 \times 10 = 100$. – Hans Lundmark Oct 25 '22 at 04:36

1 Answers1

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What you want to know is that if we are given two lists of numbers $$1, \dots, m$$ and $$1, \dots, n$$

Then how many ways are there to pick two numbers $(a, b)$ where $a$ is in the first list and $b$ is in the second list.

Note that if we fix the number $1$ from the first list. Then we can pair that number up with $n$ different numbers. More specifically, we would have the pairs $$(1, 1), \dots, (1, n)$$

This tells us that if fix a number $k$ from the first list, then we get $n$ different pairs $$(k, 1),\dots, (k, n)$$

Because there are $m$ numbers in the first list. And each number will give us $n$ pairs. The total number of pairs is $$\underbrace{n+\dots+n}_\text{$m$ times}=m\times n$$

In your specific example, the two lists are $$1, 2, 3, \dots, 10$$ and $$1, 2, 3, \dots, 10$$

Thus, the total number of ways to pick a pair $(a, b)$ where $a$ comes from the first list and $b$ comes from the second list is $10\times 10=100$.

This result holds in general. This is known as the principle of multiplication or the rule of product. Thus, if you have $n$ lists $$1, \dots, m_1\\ \vdots \\ 1,\dots ,m_n$$

Then the total number of ways to pick $n$ numbers where each number comes from each list is $$m_1\times \cdots \times m_n$$

Hope this helps!

Seeker
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