Help with understanding binomial coefficient/factorial notation

[Ttwnycorporation]

Hello, sorry for the low quality image. I require some help in understanding my notes.

It says n! = n(n-1)(n-2).... (n-r+1)(n-r)!
But where exactly does the r come from?? Isn't n just a positive integer??

Thank you

 

[Deleted member 4993]

Hello, sorry for the low quality image. I require some help in understanding my notes.

It says n! = n(n-1)(n-2).... (n-r+1)(n-r)!
But where exactly does the r come from?? Isn't n just a positive integer??

Thank you

r is some (any) positive integer which is less than or equal to n.

It will be defined by the problem.

 

[pka]

It says n! = n(n-1)(n-2).... (n-r+1)(n-r)!
But where exactly does the r come from?? Isn't n just a positive integer?

Lets take an example \(\displaystyle 10!=10\cdot 9\cdots 5\cdot 4!\) here \(\displaystyle r=4\)
It is convenient in calculation \(\displaystyle \dfrac{20!}{15!}=20\cdot 19\cdot 18\cdot 17\cdot 16\)

 

[Steven G]

Hello, sorry for the low quality image. I require some help in understanding my notes.

It says n! = n(n-1)(n-2).... (n-r+1)(n-r)!
But where exactly does the r come from?? Isn't n just a positive integer??

Thank you

r is defined in the text. Please read it more carefuly. It say ⁿC_r=.....
Now ⁿC_r means the number of ways of choosing r items from n items. That is the definition of r.

Now you do not even need to know what r stands for (othen than it is a non-negative integer. For any r<n it is true that n! = n(n-1)(n-2).... (n-r+1)(n-r)!. If you do not believe that then replace (n-r)! with (n-r)(n-r-1)(n-r-2)...3*2*1.
Now n(n-1)(n-2).... (n-r+1)(n-r)! = n(n-1)(n-2).... (n-r+1)(n-r)(n-r-1)(n-r-2)...3*2*1 and this is n! as all numbers from 1 to n (or if you prefer all numbers from n to 1) are being multiplied.

 

[Ttwnycorporation]

Hi, thank you for the reply!!
Could you show me how to derive the (n-r+1) part? I do not understand why r is needed in the first place, doesn't it just complicate things more?

 

[Ttwnycorporation]

Lets take an example \(\displaystyle 10!=10\cdot 9\cdots 5\cdot 4!\) here \(\displaystyle r=4\)
It is convenient in calculation \(\displaystyle \dfrac{20!}{15!}=20\cdot 19\cdot 18\cdot 17\cdot 16\)

Hi, thanks a lot for the reply!!
But isn't 10-r = 4, r=6 in this case? I also do not understand why r is useful, as I can just use a calculator to find out the answer!?
Thank you

 

[Ttwnycorporation]

r is some (any) positive integer which is less than or equal to n.

It will be defined by the problem.

Hi, thank you for the reply!!
The thing is, I do not understand why/how you would want/get (n-r)! as part of the expression n!


Why would you want it to 'terminate' with (n-r)!
Is (n-r) the lowest possible integer or something?

Excuse my poor math language, thank you!!

 

[pka]

But isn't 10-r = 4, r=6 in this case? I also do not understand why r is useful, as I can just use a calculator to find out the answer!?

You seem to be confused by trying to apply a formula rather than understanding process.
\(\displaystyle \dfrac{10!}{4!}=10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\)
Where that is important is in permutations: \(\displaystyle ^N\mathcal{P}_j=\dfrac{N!}{(N-j)!}\)
Thus \(\displaystyle ^{10}\mathcal{P}_4=\dfrac{10!}{(10-4)!}=10\cdot 9\cdot 8\cdot 7\)

 

[Deleted member 4993]

Hi, thanks a lot for the reply!!
But isn't 10-r = 4, r=6 in this case? I also do not understand why r is useful, as I can just use a calculator to find out the answer!?
Thank you

Suppose:

n = 123456789

r = 123456788

can you use calculator to calculate [(n!)/(r!)]?

 

[Steven G]

Suppose:

n = 123456789

r = 123456788

can you use calculator to calculate [(n!)/(r!)]?

Don't they have very BIG calculators to this these type of problems?