Permutations help

[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 dont even know what this question is asking. this is a sort of handout of 3 questions our teacher gave us in which we havent ever done any questions like this, its to challenge us, but we also get marked on it, but im drawing blanks.

can i get any sort of start off help? or atleast an explanation on what im 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. ​