Difficulty with permutation and combination

[Taylor_1989]

Right I am having an issue with the proof to permutation, I really can see the $n-r-1$
I think the confusion stems because it is in the general term, which throws me a bit, if possible could someone maybe write it in numbers and the underneath write in the general term if not too much trouble. The reason I ask for this is I am trying to understand the binomial expansion, and I have never done permutation or comnations be for, I do understand factorials and how permutation work and combination, but can't get my head around the proof for the formula.

I would like to thank anyone in advance for posting a replie to this post, much appreciated.

 

[Taylor_1989]

Okay so I have resized I am being quite vague, when I posted this. So I have been having a look at the proof a going over it so please correct me if I am wrong to what I am about to write: if I have say 5 letter - ABCDE and have 5 place to fill this I would get this type of equation: $5*4*3*2*1= 120$ So I have 120 permutation. In general form this would N=5 and the R=5 $n*n-1*n-2*n-r+1= 120$. So I am under the assumption that the $n-r+1$ is the last term in the sequence so to speak, correct or not?

 

[MarneMath]

I think you are getting it, but, I'm going to go over an informal proof to see if you can follow.

Let us select elements of S in any order. There are n elements within S, for our first selection we have n options. Thus there are n - 1 elements left in S, so we have n-1 for the second choice, and thus we now have n - 2 options left, and henece n - 2 choices for the third option. If we notice this pattern, we can say that for the rth choice there are n-(r-1) possible choices.

*So for example, when we had two choices, we had n-1 = n-(2-1)

So since each choice is indepedent we use the product rule and we obtain n(n-1)...(n-r+1) = n!/(n-r)!

Hope that clears up the last term a bit.

 

[Taylor_1989]

Yep, I see where you are coming from. I had the right idea i my head, but couldn't get what I wanted to say on here. Thanks for the input def cleared things up.