Mixed factorials: I have 16 rows and 256 columns. Values in the rows are from 1 to 2

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PTM

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Hi to FreeMathHelp!

I would like to know that how to calculate that how may combinations of values could I have in my spreadsheet.

I have 16 rows and 256 columns.

Values in the rows are from 1 to 256.

In one row I can have 256! of combinations.

In the 16 rows I can have (256!) ^ 16

What happens when I put all 16*256 values in one row with 16*256 cells and mix them?
How to calculate the possible combinations?

Thank you!
 
PTM said:
I have 16 rows and 256 columns.

Values in the rows are from 1 to 256.

In one row I can have 256! of combinations.
Click to expand...
What do you mean by having 256! "of" combinations "in one row"? Thank you! ;)
 
stapel said:
What do you mean by having 256! "of" combinations "in one row"? Thank you! ;)
Click to expand...
In the row I have 2 numbers: 1 and 2
"Mixing" them I can have (2!) > 2 combinations: 1,2 and 2,1

Having 2 rows, how many combinations of the numbers can I have in the 2 rows?
(2!)^2=4
Row 1: 1,2 Row 2: 1,2
or
Row 1: 1,2 Row 2: 2,1
or
Row 1: 2,1 Row 2: 1,2
or
Row 1: 2,1 Row 2: 2,1

What happens if I have a third row Row 3 with the values from Row 1 AND Row2
1,2,1,2 or
1,1,2,2 or
2,2,1,1 or
...
...

It's possible to count on the fingers but when the rows and columns number increasing it needs a formula to calculate it.

I'm not a student, simply interested in large numbers.

Thanks
 
Okay, well, this is all extremely poorly worded, but I think I understand what you're asking. Please make any necessary corrections, and then we'll go forward once we've established the problem statement. Best I can tell, you have a spreadsheet with 16 rows and 256 columns. In each cell there will be a number between 1 and 256. No two cells in the same row may have the same value. Accordingly, each row will have all of the numbers 1 to 256, but they can appear in any order, so there are 256! ways to arrange the numbers in each row. Then the question appears to be, how many possible spreadsheets are there. If that's the question, it's just a matter of doing the calculation of picking 16 rows from a "bucket" of 256! possible rows. You can use the standard formula given in pretty much any math textbook, or found on countless websites. I personally prefer MathWorld. The only thing left to consider is, does order matter? That is to say, in a smaller example of a 2x2 spreadsheet, are these considered unique arrangements or are they considered the same?

\(\displaystyle \begin{pmatrix}1&2\\ 2&1\end{pmatrix}\) \(\displaystyle \begin{pmatrix}2&1\\ 1&2\end{pmatrix}\)
 
PTM said:
Hi to FreeMathHelp!

I would like to know that how to calculate that how may combinations of values could I have in my spreadsheet.

I have 16 rows and 256 columns.

Values in the rows are from 1 to 256.

In one row I can have 256! of combinations.

In the 16 rows I can have (256!) ^ 16

What happens when I put all 16*256 values in one row with 16*256 cells and mix them?
How to calculate the possible combinations?

Thank you!
Click to expand...

If there is 'no restrictions' on the mix, then for one particular row there are (16*256)! ways to mix them. Continuing your example, take the third row to be
1,2,1,2
If the two 1's and two 2's are considered different [think of a red and green 1 and a read and green 2], then there would be (2*2)!=24 different mixes: (1, 1, 2, 2), (1, 1, 2, 2), etc.

Just for grins and giggles, if you are interested in large numbers, consider the following simple set of large integers
\(\displaystyle a_{1, i}\, =\, i;\, i\, \gt\, 0\)
\(\displaystyle a_{k,i}\, = i^{a_{k-1,i}};\, k\, \gt\, 1\)
A different notation was developed by Donald Knuth, that is ak,j = i^^k. The tower function, ti is the diagonal of the system, i.e.
\(\displaystyle t_i\, =\, a_{i,i}\)
which gets large fairly quickly; t5 makes the googolplex look lvery small [like basically zero], see
https://en.wikipedia.org/wiki/Googolplex

Maybe you would be interested in
http://waitbutwhy.com/2014/11/1000000-grahams-number.html
 
I agree that my post is extremely poorly worded, thank you guys for taking your time to help me.

I will return with a hopefully better described problem.
 
Ishuda said:
If there is 'no restrictions' on the mix, then for one particular row there are (16*256)! ways to mix them. Continuing your example, take the third row to be
1,2,1,2
If the two 1's and two 2's are considered different [think of a red and green 1 and a read and green 2], then there would be (2*2)!=24 different mixes: (1, 1, 2, 2), (1, 1, 2, 2), etc.
Click to expand...
No, they are not different, that's why (16*256)! cannot be true with the (16*256) cells.
I would need 4096 different values for that for the 4096 cells.
There are no red and green values, they are all black.


The numbers are actually different for the live project.

Row No2 has 65536 cells and Row No1 has to match it so it's 65536 cells as well.

The values in Row No2 can be anything randomly from 0 to 65535.
Can be a total of (65536^65536) combinations of them.
They can be swapped or simply add a value to them from Row No 1.

In Row No1 the values must be odd from 1 to 65535.
Because there's only 32768 of them, to fill all cells have to enter the 32768 values twice.
The values can be swapped only.

To make it more difficult I've chosen the prime numbers from this range, only 4096 of them.
Has to enter them 16 times to fill all 65536 cells.

Swapping cell[1*12] with cell[12*12] doesn't make a visible effect.
 
Thank you guys for all your help.


To make it more difficult I've chosen the prime numbers from this range, only 4096 of them.
Has to enter them 16 times to fill all 65536 cells.

Swapping cell[1*12] with cell[12*12] doesn't make a visible effect.
Click to expand...

If anyone in the future would like to know the solution:

65536!
__________
(16!)^4096
 
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