What are the different number of matrices available for the following

[vcakula]

Greetings,

I have a matrix of order 5 x 5

I would like to replace the

2 elements in column 1 with 0's
1 elements in column 2 with 0's
4 elements in column 3 with 0's
3 elements in column 4 with 0's
2 elements in column 5 with 0's


What are the different number of matrices available if I do replace.

Hoe can this be solved for a general case if matrix is of m x n

 

[micromass]

Depends on the original matrix. If the original matrix were

$$\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\end{array}\right)$$

then there would only be one matrix available, no matter what the replacements are.

So, you must keep track of the number of zeroes already in the matrix. For example, you can say that column i has $K_i$ zeroes.
Now, before I give any hints, what are your thoughts about the problem?? What did you attempt?

 

[vcakula]

Lets us assume that there are no 0's in the matrix and all the elements of the matrix are unique i.e there are no repetitive elements in the matrix. So the question is like replacing 2 elements in column 1 by 0's.

I tired to treat each column separately as a line with 5 places and check the number of ways 0's could be arranged.

like the
numbers of ways for arranging 2 0's be arranged in 5 places for column 1 and so on for column 2 till column 5.

then multiplied the possibilities of all the columns.

Is this correct..?

 

[haruspex]

Lets us assume that there are no 0's in the matrix and all the elements of the matrix are unique i.e there are no repetitive elements in the matrix. So the question is like replacing 2 elements in column 1 by 0's.

I tired to treat each column separately as a line with 5 places and check the number of ways 0's could be arranged.

like the
numbers of ways for arranging 2 0's be arranged in 5 places for column 1 and so on for column 2 till column 5.

then multiplied the possibilities of all the columns.

Is this correct..?

Sounds right.

 

[vcakula]

Sounds right.

Can you just solve the above problem and let me know the steps which you have followed so that I can compare my answer with yours

 

[micromass]

Can you just solve the above problem and let me know the steps which you have followed so that I can compare my answer with yours

No, we don't do that here. You need to solve the problem yourself. We will give hints or check the answer. We don't give out answers.