How to calculate permutations of a 10 x 10 matrix?

[MVFStudios]

Can anybody help me solve this problem? Here it is:

A matrix has 10 rows (labelled "A" to "J") and 10 columns (Labelled "1" to "10")

Each row represents a different feature of a car: For example, A = Maximum Speed; B = Fuel Efficiency; C = Safety, etc. An engineer designing a car can prioritize these features in order of importance from highest to lowest priority. Therefore the list of features will always consist of 10 letters, but they can be in any sequence.

Furthermore, for each of the 10 features, the engineer needs to specify "how much" of that feature will be included in the design, on a scale of 1 to 10. For example, Feature A = 8; Feature B = 5; Feature C = 9, etc. Remember that the rows with the features A to J can be ordered in any sequence, and each row can have any value between 1 and 10.

The question is, how many different 10 x 10 matrices are possible?

Thank you!

 

[tkhunny]

How many cells are there?
How many values can you put in each cell?

 

[MVFStudios]

How many cells are there?
How many values can you put in each cell?

There are in total 100 cells, i.e. 10 rows of 10 cells each. In each row you can only select one cell.

Therefore for each row there are 10 possibilities. But rows can be swapped into any sequence

 

[tkhunny]

Therefore for each row there are 10 possibilities. But rows can be swapped into any sequence

10! = 3,628,800

 

[MVFStudios]

10! = 3,628,800

Thank you! But I thought that 3,628,00 is simply the number of ways that the rows can be arranged, correct? But for each one of these arrangements each row can have a score between 1 and 10. Doesn't this increase the number?


Another way of thinking of this would be like this:

Instead of representing a different design feature, assume that each row represents a different colour. As mentioned above there are 3,628,800 ways of arranging the rows. But for each possible permutation of the rows, we are allowed to color only one cell in each row (using the color that the row represents). This would create a pattern of colored cells in the 10x10 matrix, where each row has a single colored cell. How many possible patterns can we create?