# Compute how many n-digit numbers

• knowLittle
In summary, Homework Statement Compute how many n-digit numbers can be made from the digits of at least one of {0,1,2,3,4,5,6,7,8,9 }Assume, repetition or order do not matter.The Attempt at a Solution10 choices for the 1st sub-index, 10 choices for the second sub-index, ..., 10 choices for the nth- sub-index.## 10^{n} ## total possible combinations.I think that now we need to add 'n' for a set full of one identical digit. i.e.: {2,2,...,n-th}Now
knowLittle

## Homework Statement

Compute how many n-digit numbers can be made from the digits of at least one of {0,1,2,3,4,5,6,7,8,9 }
Assume, repetition or order do not matter.

## Homework Equations

## a_{1}, a_{2}, ..., a_{n} ##

## The Attempt at a Solution

10 choices for the 1st sub-index, 10 choices for the second sub-index, ..., 10 choices for the nth- sub-index.
## 10^{n} ## total possible combinations.
I think that now we need to add 'n' for a set full of one identical digit. i.e.: {2,2,...,n-th}
Now, n*9 for all possibilities

Now, I need to take some n_i number into account in pairs and each of the n-2 numbers repeated for each number.
So, groups of two repeated for each i-th number?
Also, then I would extend this to include triple identical numbers and the rest (n-3) numbers in the set?
And, so on...?

I am sorry, if this does not make any sense or it is too messy.
Could someone give me any guidance?
Thank you.

knowLittle said:

## Homework Statement

Compute how many n-digit numbers can be made from the digits of at least one of {0,1,2,3,4,5,6,7,8,9 }
Assume, repetition or order do not matter.

How can order not matter for numbers?

## Homework Equations

## a_{1}, a_{2}, ..., a_{n} ##

## The Attempt at a Solution

10 choices for the 1st sub-index,

Is it OK to have a string of 0's for leading digits?

10 choices for the second sub-index, ..., 10 choices for the nth- sub-index.
## 10^{n} ## total possible combinations.
I think that now we need to add 'n' for a set full of one identical digit. i.e.: {2,2,...,n-th}
Now, n*9 for all possibilities

Now, I need to take some n_i number into account in pairs and each of the n-2 numbers repeated for each number.
So, groups of two repeated for each i-th number?
Also, then I would extend this to include triple identical numbers and the rest (n-3) numbers in the set?
And, so on...?

I don't follow what you are doing in this last section. Is there something about the problem you haven't told us?

The question is for a Theoretical Computer Science class.
LCKurtz said:
How can order not matter for numbers?
You might be right, but he said that it does not matter. My professor was very vague in posing the question. I asked it twice.
He gave some examples:
## {0, 0, 0,0, ...,0_{n} }={0} ##
## {0, 0, 0, ..., 0, 1_{n}}={0,1}##

LCKurtz said:
Is it OK to have a string of 0's for leading digits?
Yes, it is.
This counts as a valid number.
## { a_{0}, a_{1}, ..., a_{n} } \equiv {0_{0},0_{1},...,0_{n} } ##

LCKurtz said:
I don't follow what you are doing in this last section. Is there something about the problem you haven't told us?
Sadly, I have told you everything that was given to me.

In the last section, I wanted to start counting numbers as if repetition mattered, but I recalled that the professor said that they do not.

I will for sure ask for more clarity next time, I see him.

OK. Since leading zeroes are OK it seems to me that your original calculation of ##10^n## should do it.

1 person
Let me get back to you in a few days, I believe I can obtain more specific information about the problem. Thank you for your help so far.

It turns out that order does matter after all. One 'gotta' love language.
An example helped to elucidate.

Example:
{0,1,2,3,...,8,9, ...<whatever>,...} is a vector of size n and 1 solution that satisfies the constraints.
{3,4,7,...,9,2,1,...<whatever>,... } is another vector of size n and another solution.

Let me work it out. If you have any leads, they are welcome.
Thanks :>

Please, check the solution in attachment.
Apparently, it is incorrect. Can someone verify?
I think that I am not taking into account cases such as {m, o ,m } or {1,0,1}, where there could be repetitions.

The solution should be in the form:
Order matters
{ ... , <at least digits from 0 to 9>,..., <any numbers>, ... }

Thank you.

#### Attachments

• theorCompScie.gif
8.6 KB · Views: 491
So, here is the solution.

## \dfrac{ |n| !}{ |n_{1}|! |n_{2}|!... |n_{k}|!} \text{, where n is the size of the vector and the values in denominator are types of symbols in n that repeat.}## ##\text{For instance, if I have a vector called v={e, i, g, e, n, v, a, l, u, e}, there are say n1 type symbols.}## ##\text{n1 relates to, say, e and our |n1|=3. The numerator in the equation takes care of all combination including repetitions}## ##\text{and the denominator takes care of cases as the one just mentioned. } ##

## What is meant by "n-digit numbers"?

"N-digit numbers" refers to numbers that have a specific number of digits, where "n" represents any positive integer. For example, a 3-digit number would be any number between 100 and 999.

## What is the formula for computing how many n-digit numbers there are?

The formula for computing how many n-digit numbers there are is 10^n, where n represents the number of digits. For example, for 3-digit numbers, the formula would be 10^3 = 1000 possible numbers.

## Why is the formula for computing n-digit numbers 10^n?

This formula is based on the fact that there are 10 possible digits (0-9) for each place value in a number. So for each additional digit, there are 10 times as many possible numbers. This can be visualized with a tree diagram, where each branch represents a different digit that can be added to the number.

## What is the difference between n-digit numbers and numbers with n digits?

The difference between n-digit numbers and numbers with n digits is that n-digit numbers refers to a specific type of number (e.g. 3-digit numbers), while numbers with n digits refers to any number that happens to have n digits (e.g. 123, 4567, 89012).

## Can you give an example of computing how many n-digit numbers there are?

Sure, let's say we want to compute how many 4-digit numbers there are. Using the formula 10^n, we get 10^4 = 10,000 possible numbers. So there are 10,000 different 4-digit numbers that can be formed using the digits 0-9.

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