Proving Permutations for Natural Numbers n and r: A Comprehensive Guide

[LampMan]

Homework Statement

prove the following natural numbers n and r.

P(n-1,2) + 3P(n+1,2) = 2(2n^2 + 1) and P(n,r) = P(n-3,r-3)

The Attempt at a Solution

i honestly don't even know what this question is asking. this is a sort of handout of 3 questions our teacher gave us in which we haven't ever done any questions like this, its to challenge us, but we also get marked on it, but I am drawing blanks.

can i get any sort of start off help? or atleast an explanation on what I am trying to achieve

 

[tiny-tim]

Hi LampMan!

(try using the X² tag just above the Reply box )

prove the following natural numbers n and r.

P(n-1,2) + 3P(n+1,2) = 2(2n^2 + 1) and P(n,r) = P(n-3,r-3)

If P(n,r) is the number of ways of chooosing r objects out of n, then the first equation is fairly easy to prove.

But I don't know what the second equation is supposed to be … are you sure you have copied it correctly?

 

[Martin Rattigan]

tiny-tim wrote:

"If P(n,r) is the number of ways of chooosing r objects out of n, then the first equation is fairly easy to prove."

I think P(n,r) is meant to be the number of permutations of r objects taken from n different objects (written $^nP_r$ when I was at school), rather than the number of ways of choosing r objects from n different objects, $^nC_r$, the difference being that each different order of the r selected objects is counted as a different permutation, whereas the order is not relevant for a choice.

If P(n,r) were taken to mean $^nC_r$ as tiny-tim suggestes, the right hand side of the first equation would be double the correct value.

Either way the second equation is invalid. Mabe it should read P(n,r) $\geq$ P(n-3,r-3).

 

[tiny-tim]

Hi Martin!

…I think P(n,r) is meant to be the number of permutations of r objects taken from n different objects (written $^nP_r$ when I was at school), rather than the number of ways of choosing r objects from n different objects, $^nC_r$, the difference being that each different order of the r selected objects is counted as a different permutation, whereas the order is not relevant for a choice.

Yes, you're right, I should have been more precise …

C is the number of ways of choosing in which the order doesn't matter, and P is the number of ways of choosing in which the order matters.

Thanks for the correction. ​