Derangement and Use of -Exclusion-Inclusion Principle. (1 Viewer)

[mamun11]

Given n different objects,they have to be arranged in such a way that k objects won't occupy their initial position.How to solve this problem? For example,if n=7,k=3 how many permutations will we get so that exactly 3 objects won't occupy their initial position?

 

[aDimitri]

Given n different objects,they have to be arranged in such a way that k objects won't occupy their initial position.How to solve this problem? For example,if n=7,k=3 how many permutations will we get so that exactly 3 objects won't occupy their initial position?

i'm assuming you mean arranged in a line. if they are all different objects, doesn't that mean it's just (n! - 1) ?
Since there is only one initial position for each object.

Or is the question asking it such that NONE of the 3 objects occupy their initial position?

 

[Carrotsticks]

To do this intuitively, get the total number of ways and then subtract all the ways that people can occupy their original postilion.

The total is n!

Now subtract all the cases where one person has the right position.

But we've subtracted too many, so let's add back the number of ways that two people can be seated correctly.

Now we've added too much, now let's subtract the cases when three people occupy their original position.

Try to make a general formula from here. I'll check to see if it's right.

 

[Carrotsticks]

i'm assuming you mean arranged in a line. if they are all different objects, doesn't that mean it's just (n! - 1) ?
Since there is only one initial position for each object.

Or is the question asking it such that NONE of the 3 objects occupy their initial position?

Yes, none of them have their original position.

 

[dunjaaa]

This is what I came up with For n=7, k=3, total arrangements is = 1680

 

[Carrotsticks]

Correct