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.

 

[CellCoree]

Dear [Name],

I understand that you are having difficulty with understanding the proof for permutation and combination. It can be challenging to grasp these concepts, especially if you have not encountered them before. However, I am here to help you understand the formula and its proof.

Firstly, let's start with the general term for permutation, which is nCr, where n is the total number of objects and r is the number of objects being chosen. This can also be written as n!/(r!(n-r)!), where n! represents the factorial of n, meaning n multiplied by all the numbers below it until 1.

Now, let's break down the formula into numbers. Let's say we have 5 objects and we want to choose 3 of them. This would mean that n = 5 and r = 3. So the formula would be 5!/(3!(5-3)!), which simplifies to 5!/(3!2!), which further simplifies to (5x4x3x2x1)/((3x2x1)(2x1)). This gives us a final answer of 10, which is the total number of ways we can choose 3 objects from a set of 5.

Similarly, for combination, the formula is nCr, but this time it is n!/(r!(n-r)!), where n is the total number of objects and r is the number of objects being chosen without any order. Using the same example as before, if we want to choose 3 objects from a set of 5 without any order, the formula would be 5!/(3!(5-3)!), which simplifies to 5!/(3!2!), and finally gives us an answer of 10.

I hope this explanation helps you understand the formula better. If you have any further questions, please do not hesitate to reach out to me. Remember, practice makes perfect, so keep practicing and you will become more comfortable with permutation and combination.

Best,