Calculating Possible Combinations in 2 Vectors

[King]

Hi all,

I am working on a programming project and want to know how many possible combinations can exist in a particular situation, but I have no idea how to work it out. It isn't homework, I just want to know because I am sure it is a VERY large number and want to make a point.

There are 2 vectors, one vector is one dimensional and is, say, 5 in length with values in it: [ 1 | 2 | 3 | 4 | 5 ] and the second vector is two dimensional and empty (to be filled with the numbers in the one dimensional vector), say like so:
+---------+
| - | - | - |
| - | - | - |
| - | - | - |
+---------+
How would I go about working out working out how many combinations of those 5 numbers can be placed in the above two dimensional vector?
eg. one would be:
+---------+
| 1 | 2 | 3 |
| 4 | 5 | - |
| - | - | - |
+---------+
another:
+---------+
| 1 | 3 | 2 |
| 4 | 5 | - |
| - | - | - |
+---------+

Thank you for any help

 

[pbandjay]

There are 9 spots and 5 numbers to place in them. First pick 5 of the spots and then calculate the permutation of the 5 numbers in those 5 spots. I think this should be the answer, unless I am misunderstanding completely.

\binom{9}{5} 5! = 15,120

This can also be calculated as the number of 5-permutations of 9 objects, since the first number would have 9 possible spots to occupy, the second would have 8 possibilities, etc...

P(9,5) = 15,120

 

[King]

Thanks for the quick reply! But what are permutations? I see that the 120 came from the 5! but I'm not sure what P(9,5) is...

 

[CRGreathouse]

Thanks for the quick reply! But what are permutations? I see that the 120 came from the 5! but I'm not sure what P(9,5) is...

P(a, b) is the answer to your question with 9 slots in the matrix and 5 in the vector. It was defined on the spot, it's not a usual symbol.

 

[pbandjay]

The intermediate step is P(9,5) = 9*8*7*6*5 = 15,120.

Permutations are similar to combinations except in permutations, the order matters. In combinations, the order does not.

 

[King]

Thank you very much for the help